GIFT  OF 
Professor  Robertson 


ENGINEERING  LIBRARY 


/  V 


ex,  , 


PRINCIPLES 

OF 

ELECTRICAL  MEASUREMENTS 


McGraw-Hill  BookGompaiiy 


Electrical  World         TTie  Engineering  andMining  Journal 
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PRINCIPLES 

OF 

ELECTRICAL  MEASUREMENTS 


BY 
ARTHUR  WHITMORE  SMITH,  PH.  D. 

ASSOCIATE  PROFESSOR  OF  PHYSICS,  UNIVERSITY  OF  MICHIGAN 


FIRST  EDITION 
SECOND  IMPRESSION 


McGRAW-HILL  BOOK  COMPANY,  INC. 
239  WEST  39TH  STREET,  NEW  YORK 

6  BOUVERIE  STREET,  LONDON,  E.  C. 

1914 


\A-        r\ 


COPYRIGHT,  1914,  BY  THE 
MCGRAW-HILL  BOOK  COMPANY,  INC. 


•I  FT  OF 


\    \6 

Engineering 
Library 


THE .  MAPLE . PKE88*  YORK .  PA 


PREFACE 

This  book  is  written  for  the  instruction  of  those  who  are 
beginning  their  course  in  Electrical  Engineering,  or  who  desire 
a  more  complete  understanding  of  this  branch  of  Physics  than 
is  afforded  in  most  elementary  manuals,  and  as  far  as  possible 
the  first  consideration  has  been  the  requirements  of  such 
readers.  It  is  the  result  of  ten  years  of  teaching  the  subject 
in  the  University  of  Michigan  and  as  now  presented  it  meets 
the  requirements  for  a  class  book  as  well  as  a  laboratory 
manual.  The  aim  has  been  to  lead  the  student  to  learn  the 
facts  from  his  own  observations,  and  direct  information  is  often 
replaced  by  a  suggestion  how  the  information  can  be  obtained. 
This  leads  to  independent  investigation  and  does  not  dull  the 
keen  pleasure  of  discovery  by  knowing  the  result  before  the 
experiment  is  tried. 

The  book  is  arranged  on  the  progressive  system.  The  sim- 
pler and  more  fundamental  parts  of  the  subject  are  taken  up  in 
the  first  chapters,  and  in  the  first  part  of  each  chapter,  while  the 
more  difficult  measurements  and  the  methods  involving  more 
extended  knowledge  are  reserved  until  the  student  has  attained 
greater  proficiency.  For  example,  Chapter  I  shows  how  to 
measure  current,  E.M.F.,  resistance,  and  power,  by  ammeter 
and  voltmeter  methods.  For  an  elementary  course  in  Elec- 
trical Measurements  nothing  could  be  better  than  this  series 
of  simple  experiments,  well  understood.  They  bring  out  the 
fundamental  relations  with  a  minimum  of  apparatus  to  confuse 
the  mind;  and  they  are  not  out  of  place  at  the  beginning  of  a 
more  extended  course  which  contemplates  using  the  entire 
book. 

In  deducing  the  formulas  it  has  not  been  considered  sufficient 
merely  to  write  down  the  equations,  but  in  each  case  they  are 


868522 


vi  PREFACE 

worked  out  logically  from  the  fundamental  relations,  and  the 
reader  is  led  to  use  his  reason  rather  than  his  memory.  Thus 
in  Chapter  VII,  considerable  use  is  made  of  KirchhofFs  second 
law,  not  so  much  on  account  of  the  importance  of  the  law  itself 
but  because  it  enables  one  to  write  down  in  the  form  of  equa- 
tions many  of  the  relations  that  are  found  to  exist  when  the 
experiment  is  performed  in  the  laboratory.  There  is  only 
enough  repetition  to  make  the  work  intelligible  at  whatever 
point  it  may  be  taken  up,  as  must  be  necessary  in  a  large  class 
where  all  cannot  take  the  experiments  in  the  order  in  which 
they  are  given  in  the  book. 

As  the  subject  is  carried  further  more  theoretical  treatment 
is  required,  until  in  the  chapters  on  magnetism  the  relations 
between  the  quantities  involved  had  to  be  very  completely 
worked  out  from  the  fundamental  definitions  because  this  part 
of  the  subject  is  lightly  passed  over  by  most  writers.  In  fact, 
the  ideas  of  many  students  regarding  B  and  H  are  not  only 
vague,  but  often  are  erroneous,  and  it  has  been  my  privilege 
to  lead  several  hundred  such  students  to  a  clear  and  useful 
understanding  of  this  interesting  part  of  Physics.  If  this  book 
shall  be  the  means  of  enlarging  the  circle  of  those  who  are  thus 
helped  to  think  for  themselves  I  shall  feel  that  it  has  not  been 
written  in  vain. 

I  desire  to  express  my  thanks  to  Dean  Karl  E.  Guthe, 
who  has  kindly  read  the  entire  proof,  and  to  Assistant  Pro- 
fessor Neil  H.  Williams,  who  has  read  the  chapters  on  mag- 
netism and  alternating  currents;  also  to  my  wife,  Frances 
Berry  Smith,  for  assistance  in  the  preparation  of  the  index. 

ARTHUR  WHITMORE  SMITH. 
UNIVERSITY  OF  MICHIGAN, 
Sept.,  1914. 


CONTENTS 

PREFACE PAGE  v 

INTRODUCTION 
UNITS  AND  DEFINITIONS 

1.  C.  G.  S.  Systems 1 

2.  Unit  Magnetic  Pole 1 

3.  Unit  Magnetic  Field 2 

4.  Magnetic  Effect  of  an  Electric  Current 3 

5.  Unit  Quantity 4 

6.  Resistance 5 

7.  Difference  of  Potential,  Electromotive  Force,   Fall  of 

Potential 5 

8.  Unit  Potential  Difference » 6 

9.  The  Practical  Units 6 

10.  Concrete  Examples  of  these  Units 7 

11.  The    Conference   on   Electrical   Units   and   Standards. 

London,  1908.    . 7 

CHAPTER  I 
AMMETER  AND  VOLTMETER  METHODS 

12.  Laws  of  Electric  Currents — Use  of  an  Ammeter.    ...  10 

13.  Fall  of  Potential  in  Electric  Circuit— Use  of  a  Voltmeter  12 

14.  Ohm's  Law. 14 

15.  Measurement  of  Resistance  by  Ammeter  and  Voltmeter .  15 

16.  Measurement  of  Resistance  by  Ammeter  and  Voltmeter  16 

17.  To  Find  the  Best  Arrangement  for  Measuring  Resistance 

with  an  Ammeter  and  a  Voltmeter 17 

'-•"  18.     Internal  Resistance  of  a  Battery 19 

19.  A  More  Exact  Method 20 

20.  Relation  between  Available  E.M.F.  and  Current    ...  21 

21.  Useful  Power  from  a  Cell 23 

22.  E.M.F.  of  a  Cell  by  a  Voltmeter  and  an  Auxiliary 

Battery 25 

vii 


viii  CONTENTS 

23.  Measurement  of  Current  by  a  Voltmeter  and  Shunt .    .  PAGE     27 

24.  Measurement  of  a  High  Resistance  by  a  Voltmeter  Alone  28 

25.  Time  Test  of  a  Cell 29 

CHAPTER  II 
BALLISTIC  GALVANOMETER  AND  CONDENSER   METHODS 

26.  Capacity. — Elementary  Ideas 32 

27.  Condensers 33 

28.  Unit  Capacity 34 

29.  Ballistic  Galvanometer 34 

30.  Use  of  Ballistic  Galvanometer  and  Condenser     ....  35 

31.  Damping  of  a  Galvanometer.     Critical  Damping  ...  36 

32.  The  Constant  of  a  Ballistic  Galvanometer 37 

33.  Comparison  of  E.M.F.'s  by  Condenser  Method  ....  39 

34.  Comparison  of  Capacities  by  Direct  Deflection  ....  40 

35.  Internal  Resistance  of  a   Battery  by  the   Condenser 

Method 41 

36.  Insulation  Resistance  by  Leakage      44 

37.  Insulation  Resistance  by  Leakage      45 

CHAPTER  III 
THE  CURRENT  GALVANOMETER 

38.  Description  of  a  Galvanometer 47 

39.  Figure  of  Merit 48 

40.  Figure  of  Merit 49 

41.  Other  Constants  of  the  Galvanometer 50 

42.  Use  of  Shunts 51 

43.  Universal  Shunt 52 

44.  The  Multiplying  Power  of  a  Shunt 54 

45.  Resistance  of  Galvanometer  by  Half  Deflection      ...  55 
1    46.     Resistance  of  a  Galvanometer  by  Half  Deflection  ...  56 

47.  Differential  Galvanometer 56 

47a.  Differential  Galvanometer  in  Shunt 58 

CHAPTER  IV 
THE  WHEATSTONE  BRIDGE 

48.  The  Wheatstone  Bridge 61 

:    49.     The  Slide  Wire  Bridge.— Simple  Method 62 


CONTENTS  ix 

50.  Calibration  of  the  Slide  Wire  Bridge PAGE    64 

51.  Double  Method  of  Using  the  Slide  Wire  Bridge     ...  65 

52.  The  Wheatstone  Bridge  Box 66 

53.  Location  of  Faults 69 

54.  Methods  for  Locating  Faults 71 

55.  The  Murray  Loop 71 

56.  Fisher's  Method 72 

57.  Location  of  a  Cross 73 

58.  Location  of  a  Cross • 74 

59.  Location  of  Opens 75 

60.  Resistance  of  Electrolytes 75 

CHAPTER  V. 
THE  WHEATSTONE  BRIDGE  (Continued) 

61.  The  Slide  Wire  Bridge  with  Extensions .  78 

62.  To  Find  the  Length  of  the  Bridge  Wire  with  Its  Ex- 

tensions    80 

63.  To  Calibrate  the  Slide  Wire  Bridge  with  Extensions      .  81 

64.  Advantages  of  the  Double  Method 82 

65.  The  Best  Position  of  Balance 84 

66.  Sources  of  Error  in  Using  the  Slide  Wire  Bridge.    .   ,  86 

67.  The  Direct  Reading  Bridge 86 

68.  Measurement  of  Resistance  by  Carey  Foster's  Method  87 

69.  To  Determine  the  Value  of  p 89 

70.  Temperature  Coefficient  of  Resistance      89 

CHAPTER  VI 
MEASUREMENT  OF  CURRENT 

71.  Hot  Wire  Ammeter 92 

72.  The  Weston  Ammeter      92 

73.  The  Weston  Voltmeter 93 

74.  Galvanometers 93 

75.  The  Tangent  Galvanometer 94 

76.  The  Coulometer 95 

77.  The  Kelvin  Balance 96 

r  78.     The  Electrodynamometer 98 

79.  Calibration  of  an  Electrodynamometer 99 

80.  Measurement  of  Current  by  Standard  Cell      99 


x  CONTENTS 

CHAPTER  VII 
POTENTIOMETER  METHODS 

81.  Potential  Difference PAGE  100 

82.  Kirchhoff's  Two  Laws      100 

83.  Illustrations  of  Kirchhoff's  Second  Law 102 

84.  Proof  of  Kirchhoff's  Second  Law 103 

85.  The  Potentiometer  Method 104 

.     86.     The  Resistance  Box  Potentiometer .  "  106 

87.  The  Potentiometer 108 

88.  Standard  Cells 109 

89.  The  Weston  Standard  Cell 110 

90.  Use  of  a  Standard  Cell 110 

>     91.     Comparison  of  Resistances  by  the  Potentiometer   .    .    .  Ill 

(    92.     Calibration  of  a  Voltmeter 112 

1    93.     A  More  Convenient  Method 114 

94.  Calibration  of  a  Low  Reading  Voltmeter 114 

95.  Measurement  of  Current  by  Means  of  a  Standard  Cell .  115 
l    96.     Calibration  of  an  Ammeter *.    .    .  116 

97.     Calibration  of  a  Mil-ammeter 117 

CHAPTER  VIII 
MEASUREMENT  OF  POWER 


98.  The  Measurement  of  Electrical  Power      119 

99.  The  Use  of  an  Electrodynamometer  for  the  Measure- 

ment of  Power 119 

100.  The  Weston  Wattmeter 121 

101.  Comparison  of  a  Wattmeter  with  an  Ammeter  and  a 

Voltmeter 122 

102.  Power  Expended  in  a  Rheostat 123 

102a.  Efficiency  of  Electric  Lamps 124 

103.  Measurement  of  Power  in  Terms  of  a  Standard  Cell     .  125 

104.  Calibration  of  a  Non-compensated  Wattmeter    ....  127 

105.  Calibration  of  a  Compensated  Wattmeter 129 

106.  Calibration  of  a  High  Reading  Wattmeter 132 


CONTENTS  xi 

CHAPTER  IX 
MEASUREMENT  OF  CAPACITY 

107.  Laws  of  Condensers PAGE  134 

108.  Comparison  of  Capacities  by  Direct  Deflection  ....  135 

109.  Bridge  Method  for  Comparing  Two  Capacities  ....  136 

110.  Comparison  of  Capacities  by  Gott's  Method 138 

111.  Comparison  of  Capacities  by  the  Method  of  Mixtures  .  139 

112.  Study  of  Residual  Discharges 142 

113.  The  Ballistic  Galvanometer 143 

114.  Turning  Moment  Due  to  a  Current 143 

115.  To  Express  co  in  Terms  of  Known  Quantities      ....  144 

116.  Period  of  Oscillation 146 

117.  Correction  for  Damping 147 

118.  Final  Formulas 149 

119.  Absolute  Capacity  of  a  Condenser 149 

CHAPTER  X 
THE  MAGNETIC  CIRCUIT 

120.  Introduction 151 

121.  The  Magnetic  Circuit 151 

122.  Magnetic  Flux 152 

123.  Magnetomotive  Force 153 

124.  Reluctance 154 

125.  Flux  from  a  Permanent  Magnet 155 

126.  Study  of  a  Magnetic  Circuit— Bar  and  Yoke 156 

127.  Computation  for  Magnetomotive  Force 157 

128.  Measurement  of  Magnetic  Flux 157 

129.  To  Find  the  Constant  c 157 

130.  Plotting  the  Results 158 

131.  Standard  Curve— Unit  Circuit 158 

CHAPTER  XI 
DEFINITION  OF  A  MAXWELL 

132.  Complete  Definition  of  the  Value  of  Unit  Flux  ....  160 

133.  Relations  between  Current  and  the  Magnetic  Field  .    .  160 

134.  Force  between  Current  and  Magnetic  Field     ...  161 


xii  CONTENTS 

135.  Magnetic  Induction      PAGE  164 

136.  Force  Exerted  on  a  Conductor  Carrying  a  Current    .    .  165 

137.  Work  When  a  Current  Moves  through  a  Magnetic  Field .  165 

138.  Induced  Electromotive  Force 166 

139.  Definition  of  a  Maxwell 167 

140.  Measurement  of  Magnetic  Flux 168 

141.  Relation  between  Field  Intensity  and  M.M.F     ....  168 

142.  Long  Straight  Current .  169 

143.  Ring  Solenoid 169 

144.  Long  Straight  Solenoid :    .    .    .  170 

145.  Magnetic  Field  Due  to  any  Electric  Current 170 

146.  Magnetic  Effect  of  a  Current  Element 171 

147.  Extension  to  General  Case 172 

148.  Magnetic  Field  at  Center  of  a  Circle 172 

CHAPTER  XII 
MAGNETIC  TESTS  OF  IRON  AND  STEEL 

149.  Introduction 174 

150.  Double  Bar  and  Yoke 174 

151.  To  Determine  the  Constant  of  the  Galvanometer  .    .    .  176 

152.  To  Find  the  Value  of  B.  from  the  Deflection 177 

153.  To  Determine  the  Value  of  H 177 

154.  Permeability 178 

155.  The  Ring  Method 178 

156.  The  Step  by  Step  Method 179 

157.  Hysteresis— Step  by  Step  Method 180 

*"  158.  Hysteresis  by  Direct  Deflection 181 

159.  Determination  of  the  Values  of  B      184 

160.  The  Magnetic  Ballistic  Constant,  c 185 

161.  The  Values  of  the  Magnetizing  Force,  H 185 

162.  Energy  Loss  through  Hysteresis 185 

163.  Permanent  Magnets 186 

CHAPTER  XIII 
ELECTROMAGNETIC  INDUCTION 

164.  Electromagnetic  Induction      187 

165.  Laws  of  Mutual  Induction 187 

166.  The  Effect  of  Varying  the  Primary  Current 188 

167.  The  Effect  of  Varying  the  Secondary  Resistance    ...  189 


CONTENTS  xiii 

168.  Meaning  of  Mutual  Inductance PAGE  191 

169.  Calculation  of  the  Value  of  M 193 

170.  To   Determine  the  Constant  of  the  Ballistic    Galvan- 

ometer   194 

171.  Constant  of  a  Ballistic  Galvanometer  by  Means  of  a 

Coil  of  Known  Mutual  Inductance 195 

172.  Meaning  of  Self  Inductance 197 

173.  Starting  and  Stopping  a  Current 197 

174.  Dying  Away  of  a  Current 198 

175.  Beginning  of  a  Current 199 

CHAPTER  XIV 
MEASUREMENT  OF  SELF  AND  MUTUAL  INDUCTANCE 

176.  Comparison  of  Two  Self  Inductances 202 

177.  Comparison  of  a  Mutual  Inductance  with  a  Self  Induct- 

ance   205 

178.  Measurement  of  a  Self  Inductance  by  Means  of  a  Capac- 

ity.    Maxwell's  Method 206 

179.  Case  A.    Self  Inductance  Varied 208 

180.  Case  B.    Resistance  Varied 208 

181.  Case  C.    Capacity  Varied 209 

182.  Case  D.    Effect  of  Capacity  Varied 210 

183.  Anderson's  Method  for  Comparing  a  Self  Inductance 

with  a  Capacity 210 

184.  Direct  Comparison  of  Two  Mutual  Inductances .    .    .    .  212 

185.  Comparison  of  Two  Unequal  Mutual  Inductances.    .    .  213 
185a.  Comparison   of  a  Large    Mutual   Inductance  with  a 

Small  One 215 

186.  Measurement  of  a  Mutual  Inductance  in  Terms  of  a 

Known  Capacity.     Carey  Foster's  Method 217 

CHAPTER  XV 
ALTERNATING  CURRENT 

187.  An  Alternating  Current 220 

188.  Tracing  Alternating-current  and  E.M.F.  Curves  .    .    .  220 

189.  Measurement  of  an  Alternating  Current 223 

190.  Definition  of  an  Ampere  of  Alternating  Current.    .    .    .  224 

191.  Average  Value  of  a  Sine  Current 224 

192.  Mean  Square  Value  of  a  Sine  Current 225 


xiv  CONTENTS 

193.  To  Find  what  E.M.F.  is  Required  to  Maintain  a  Given 

Current PAGE  226 

194.  Graphical  Solutions 227 

195.  Mean  Square  Values 228 

196.  Measurement  of   Impedance   by  Ammeter  and    Volt- 

meter    228 

197.  Self  Inductance  by  the  Impedance  Method 229 

198.  Impedance  and  Angle  of  Lag  by  the  Three  Voltmeter 

Method 229 

199.  Determination  of  Equivalent  Resistance 231 

200.  Inductive  Circuit  in  Series 231 

201.  Inductive  Circuits  in  Parallel 233 

202.  Graphical  Solution  for  Circuits  Having  Capacity   ....  234 

203.  Power  Expended  in  an  Alternating-current  Circuit    .    .  237 

INDEX  239 


PRINCIPLES  OF 
ELECTRICAL  MEASUREMENTS, 

INTRODUCTION  /, :    ; ;;  ;  ,  £  fc  ; 

UNITS  AND  DEFINITIONS 

1.  C.G.S.    Systems. — In   the   measurement   of   electrical 
quantities  there  are  two  distinct  systems  of  units  that  may  be 
used.     In  the  electrostatic  system  the  fundamental  unit  is 
determined  by  the  repulsion  of  two  similar  charges  of  electric- 
ity.    In  the  electromagnetic  system  it  is  the  repulsion  of  two 
similar  magnetic  poles  that  determines  the  values  of  the  units. 

Each  of  these  systems  is  properly  called  an  " absolute," 
or  a  " C.G.S."  system,  but  all  of  the  common  units  such  as 
ohm,  ampere,  volt,  etc.,  are  based  solely  upon  the  electro- 
magnetic C.G.S.  system  of  units. 

2.  Unit  Magnetic  Pole. — When  two  bars  of  steel  are  mag- 
netized it  is  found  that  one  end  of  one  will  attract  one  end  of 
the  other,  while  the  other  end  is  repelled  by  the  same  end  of  the 
first  one.     Particularly  is  this  the  case  if  the  bars  are  long  thin 
needles  with  ball  ends. 

The  laws  relating  to  this  attraction  and  repulsion  have  been 
carefully  studied,  and  while  the  resultant  effect  must  actually 
be  the  sum  of  all  the  separate  effects  of  each  part  of  one  mag- 
net upon  each  part  of  the  other,  yet  for  purposes  of  computation 
the  correct  result  is  obtained  quite  simply  if  the  action  is 
considered  as  due  to  a  "pole"  concentrated  at  a  point  not  far 
from  the  center  of  the  ball  on  the  end  of  a  ball-ended  magnet. 
As  the  result  of  careful  measurements,  Coulomb  found  that 
the  force  of  repulsion  between  two  similar  poles  is  given  by  the 
expression, 


mm* 


F  =  k  -^-  dynes, 

where  r  is  the  distance  in  centimeters  between  the  two  points 

1 


2  ELECTRICAL  MEASUREMENTS 

at  which  are  concentrated  the  poles  of  strength  m  and  m' 
respectively. 

Faraday  further  showed  that  if  the  stationary  medium  be- 
tween these  poless.has  a  permeability  /*,  the  force  is  then  fj,  times 
smaller.  '  This  gives, 


'        -  -  •«••*  ",'r»          7  ^ 

F  =  k  —  dynes, 

as  the  complete  expression  for  the  force  between  two  poles; 
k  is  the  proportionality  factor  to  give  the  result  in  dynes. 

Apparently  the  most  rational  unit  with  which  to  measure 
pole  strength  would  be  one  of  such  a  value  as  to  make  k  =  1. 

Definition.  —  A  unit  magnetic  pole  is  one  of  such  strength 
that  when  placed  1  cm.  from  an  equal  pole,  in  a  vacuum,  it  will 
be  repelled  with  a  force  of  I  dyne. 

3.  Unit  Magnetic  Field.  —  When  a  magnetic  pole  is  brought 
near  a  magnet  or  an  electric  current  it  experiences  certain 
forces  urging  it  to  move  in  a  definite  direction.  A  force  which 
thus  acts  upon  a  magnetic  pole  is  called  a  magnetic  force; 
and  the  region  in  which  magnetic  force  is  manifested  is  called 
a  magnetic  field. 

The  force  exerted  upon  a  magnetic  pole  depends  quite  as 
much  upon  the  strength  of  the  pole  as  upon  the  field,  being 
given  by  the  relation, 

F  =  Hm,          or          H  =  ~  . 

The  intensity  of  the  magnetic  force  in  a  magnetic  field,  or 
more  briefly,  the  field  intensity,  is  thus  measured  by  the  force 
per  unit  pole  which  is  exerted  upon  a  magnetic  pole.  The  name 
"gauss"  for  unit  magnetic  field  was  adopted  by  the  Inter- 
national Electrical  Congress  at  Paris  in  1900.1 

Definition.  —  A  gauss  is  the  intensity  of  a  magnetic  field  in 
which  a  magnetic  pole  experiences  a  force  of  1  dyne  per  unit 
pole. 

1  See  Elec.  Rev.,  Vol.  47,  p.  441,  1900. 


INTRODUCTION  3 

4.  Magnetic  Effect  of  an  Electric  Current. — When  the  poles 
of  a  battery  are  joined  by  a  wire  various  things  happen. 
Among  others,  the  wire  becomes  warm  to  a  greater  or  less 
degree  and  in  some  cases  may  even  be  melted.  More  impor- 
tant still  the  space  surrounding  the  wire  becomes  a  magnetic 
field  in  which  a  small  magnetized  needle  always  tends  to  set 
itself  at  right  angles  to  the  wire.  A  careful  examination  of  this 
field  shows  that  the  positive  pole  of  such  a  needle  is  urged 
along  a  plane  curve  encircling  the  wire,  which  for  the  case  of 
a  straight  wire  becomes  a  circle  whose  center  is  at  the  center 
of  the  wire  and  whose  plane  is  normal  to  the  wire. 

When  these  phenomena  are  observed  it  is  said  that  an  elec- 
tric current  is  flowing  along  the  wire.  The  positive  direction 
of  this  current  is  taken,  by  convention,  as  bearing  the  same 
relation  to  the  direction  of  the  force  acting  upon  the  positive 
pole  as  the  translation  of  a  right-handed  screw  bears  to  the 
direction  of  its  rotation. 

It  is  evident  that  considering  the  magnetic  needle  as  a  whole, 
there  is  no  resultant  force  tending  to  move  it  in  any  direction, 
for  whatever  forces  act  upon  one  pole,  they  are  balanced  by 
equal  and  opposite  forces  acting  upon  the  other  pole.  Thus 
the  negative  pole  of  the  magnet  is  urged  around  the  current 
in  the  opposite  direction,  and  therefore  no  useful  work  can  be 
obtained  by  carrying  the  magnet  as  a  whole  around  the  currejit. 
For  while  the  positive  pole  will  be  urged  along  with  a  force 
proportional  to  the  intensity  of  the  magnetic  field,  it  will  be 
necessary  to  supply  an  equal  force  to  carry  the  negative  pole 
along  the  same  direction.  But  for  purposes  of  definition  we 
can  consider  these  forces  separately. 

The  existence  of  these  forces  means  that  work  is  done 
whenever  the  poles  of  the  magnet  are  moved,  positive  work 
being  done  by  one  pole  and  negative  work  by  the  other.  For 
simplicity  let  us  consider  only  the  positive  pole.  The  amount 
of  work  required  to  carry  such  a  pole  once  around  the  current 
and  back  to  the  starting  point  against  the  forces  of  the  mag- 
netic field  depends  upon  the  strength  of  the  pole  and  the  value 


4  ELECTRICAL  MEASUREMENTS 

of  the  current.  When  the  work  is  measured  in  ergs  this  leads 
at  once  to  the  definition  of  the  electromagnetic  C.G.S.  unit  of 
current. 

Definition. — Unit  current  is  flowing  in  a  circuit  when  it  re- 
quires 4-7T  ergs  per  unit  pole  to  carry  a  magnetic  pole  once 
around  the  current. 

This  is  the  fundamental  definition  of  unit  current,  and  in 
addition  to  being  brief  it  will  be  found  most  useful.  One 
advantage  of  this  definition  is  that  it  defines  the  value  of  a  real 
current,  flowing  through  an  actual  circuit,  in  terms  of  a  mag- 
netic pole  of  some  possible  size.  (See  Chapters  X  and  XI  for 
further  applications  of  this  definition.) 

The  following  corollary  must  also  be  true. 

COROLLARY  I. — The  work  required  to  carry  a  magnetic 
pole  round  any  path  not  enclosing  a  current  is  zero. 

When  a  current  flows  through  a  circular  loop  of  wire,  the  only 
point  that  is  symmetrically  located  with  respect  to  the  current 
is  the  center  of  the  circle.  The  intensity  of  the  magnetic 
field  at  this  point  depends  not  only  upon  the  value  of  the  cur- 
rent, but  also  upon  the  length  of  wire  in  the  loop  and  its  dis- 
tance from  the  center.  This  leads  to 

COROLLARY  II. — A  current  of  1  C.G.S.  unit  flowing 
through  a  wire  bent  to  the  arc  of  a  circle  of  1  cm.  radius 
will  produce  at  the  center  a  field  intensity  of  1  gauss  for 
each  centimeter  length  of  wire. 

The  4  TT  enters  the  definition  above  in  order  to  give  to  this 
corollary  the  apparent  simplicity  with  which  it  is  stated.  In 
some  respects  it  would  have  been  simpler  if  the  unit  current 
could  have  been  defined  4  TT  times  smaller. 

For  proof  of  Corollary  II.  see  Chapter  XI. 

5.  Unit  Quantity. — The  definition  of  unit  quantity  of  elec- 
tricity follows  at  once  from  the  definition  of  unit  current. 

Definition. — Unit  quantity  of  electricity  is  the  quantity  trans- 
ferred by  unit  current  in  one  second. 

This  unit  quantity  is  3  X  1010  times  larger  than  the  electro- 
static unit  of  quantity  mentioned  at  the  beginning  of  page  1. 


INTRODUCTION  5 

6.  Resistance. — When,  an  electric  current  is  flowing  through 
a  circuit  it  requires  a  continual  expenditure  of  energy  to  main- 
tain the  current.     This  energy  appears  in  the  form  of  heat 
in  the  conductor  carrying  the  current.     The  amount  of  heat 
thus  generated  depends  upon  the  amount  of  current  flowing 
through  the  conductor,  the  time  that  it  flows,  and  upon  a 
property  of  the  conductor  called  its  resistance.     By  using 
different  currents  and  different  conductors  it  has  been  found 
that  the  amount  of  heat  produced  in  a  given  conductor  is 
proportional  to  the  square  of  the  current,  so  that,  if  W  denotes 
this  amount  of  heat  energy  measured  in  ergs, 

W  =  R  P  t  ergs, 

where  R  denotes  the  resistance  of  the  conductor  and  t  the  time 
that  the  current  I  is  flowing.  Of  course  the  resistance  must 
be  measured  in  units  of  such  magnitude  as  will  make  this  ex- 
pression a  true  equation.  This  leads  at  once  to  the  following 
definition: 

Definition. — Unit  resistance  is  that  resistance  in  which  one 
erg  of  heat  is  produced  each  second  by  the  passage  of  unit 
current. 

7.  Difference  of  Potential,  Electromotive  Force,  Fall  of 
Potential. — The  difference  of  potential  between  two  points  is 
that  difference  in  condition  which  produces  a  current  from 
one  point  to  the  other  as  soon  as  they  are  connected  by  a 
conductor.     Every  battery  or  other  electric  generator  possesses 
a  certain  power  of  maintaining  a  difference  of  potential  be- 
tween its  terminals,  and  therefore  a  power  of  driving  a  con- 
tinuous current.     This  difference  of  potential  produced  by  a 
cell  or  other  generator,  and  which  may  be  considered  as  the 
cause  of  the  current,  is  called  electromotive  force.     It  must  be 
remembered  that  this  quantity  is  not  a  force  at  all,  and  in  order 
to  avoid  using  the  word  "force"  it  is  commonly  called  E.M.F. 

When  a  current  flows  through  a  conductor  there  is  a  difference 
of  potential  between  any  two  points  on  the  conductor.  This 
difference  of  potential  is  greater  the  further  apart  the  points 


6  ELECTRICAL  MEASUREMENTS 

are  taken  and  as  the  change  is  gradual  it  is  usually  called  a 
"  fall  of  potential."  It  can  always  be  expressed  by  the  formula 
R  I,  where  R  is  the  resistance  of  the  conductor,  or  conductors, 
under  consideration. 

This  apparent  duplication  of  names  may  at  first  appear  un- 
necessary, but  the  corresponding  ideas  are  quite  distinct  and 
the  correct  use  of  the  proper  term  will  add  conciseness  to  one's 
thinking  and  speaking.  Thus  we  have  the  E.M.F.  of  a  battery ; 
the  fall  of  potential  along  a  conductor;  and  the  more  general 
and  broader  term,  difference  of  potential,  which  includes  both 
of  the  above  as  well  as  some  others  for  which  no  special  names 
are  used. 

8.  Unit  Potential  Difference. — Having  now  defined  the  unit 
of  current  and  the  unit  of  resistance,  the  value  to  be  chosen  for 
the  unit  difference  of  potential  follows  from  Ohm's  law.1 

Definition.  —  Unit  potential  difference  is  the  difference  of 
potential  over  unit  resistance  when  carrying  unit  current. 

Since  E  =  RI,  and  Q  =  It,  from  the  preceding  definitions, 
the  amount  of  energy  expended  in  a  conductor  by  an  electric 
current  may  be  expressed  as 

W  =  R  P  t  =  E  Q, 

where  Q  denotes  the  total  quantity  of  electricity  that  has  passed 
through  the  conductor.  This  gives  another  way  of  stating  the 
value  of  unit  potential  difference,  viz., 

COROLLARY. — Unit  difference  of  potential  is  that  difference 
of  potential  through  which  one  erg  can  raise  a  unit  quantity 
of  electricity. 

9.  The  Practical  Units.      The  Ohm. — The  unit  of  resistance 
just  defined  is  inconveniently  small  (because  an  erg  is  so  small) 
and  therefore  for  practical  use  a  multiple  unit,  1,000,000,000 
times  as  large,  is  used.     This  larger  unit  is  called  an  ohm. 

The  Ampere. — In  the  same  way,  the  unit  of  current  defined 
above  turns  out  to  be  too  large  for  convenience  in  practical 
measurements,  and  therefore  the  ampere  is  defined  as  being 
one-tenth  of  the  C.G.S.  unit. 

'See  page  14. 


INTRODUCTION  7 

The  Volt. — Having  now  defined  the  ampere  and  the  ohm,  it 
follows  that  the  practical  unit  for  difference  of  potential  is 
the  difference  of  potential  that  steadily  applied  to  a  conductor 
whose  resistance  is  1  ohm  will  produce  a  current  of  1  ampere. 
This  unit  is  called  a  volt.  Evidently  it  is  100,000,000  C.G.S. 
units. 

The  Coulomb. — Having  the  ampere  for  the  practical  unit 
of  current  it  follows  that  the  corresponding  unit  of  quantity 
is  the  quantity  transferred  by  one  ampere  in  one  second. 
This  unit  is  called  a  coulomb. 

The  Watt. — The  power  expended  in  maintaining  one  am- 
pere under  a  potential  difference  of  one  volt  is  called  a  watt. 

The  Joule. — The  work  done  each  second  by  one  watt  is 
called  a  joule.  This  is  the  work  that  will  transfer  one  cou- 
lomb through  a  potential  difference  of  one  volt. 

10.  Concrete  Examples  of  these  Units. — These  definitions 
of  the  electrical  units  do  not  furnish,  directly,  any  standards 
for  use  in  making  actual  measurements.     Therefore  the  most 
careful  investigations  have  been  made  to  determine  the  tan- 
gible values  of  these  units   as   defined   above,  in   order  to 
express  them  in  terms  of  definite  concrete  quantities.     From 
the  very  nature  of  the  case,  such  determinations  never  can 
express  the  exact  values  of  the  units,   but  they  have  been 
determined  much  closer  than  is  ever  required  in   ordinary 
measurements.     At  a  conference  of  scientific  delegates,  which 
met  in  London,  Oct.  12,  1908,  the  following  resolutions  were 
adopted.     These  resolutions  state  the  concrete  values  of  the 
fundamental  units  in  accordance  with  the  best  and   latest 
opinion  of  the  scientific  men  of  the  world. 

11.  The  Conference  on  Electrical  Units  and  Standards. 
London,  1908. 

RESOLUTIONS 

I.  The  Conference  agrees  that  as  heretofore  the  magnitudes 
of  the  fundamental  electric  units  shall  be  determined  on  the 
Review,  vol.  63,  1908,  p.  738. 


8  ELECTRICAL  MEASUREMENTS 

electromagnetic  system  of  measurement  with  reference  to  the 
centimeter  as  the  unit  of  length,  the  gram  as  the  unit  of  mass, 
and  the  second  as  the  unit  of  time. 

These  fundamental  units  are  (1)  the  Ohm,  the  unit  of 
electric  resistance  which  has  the  value  of  1,000,000,000  in 
terms  of  the  centimeter  and  second;  (2)  the  Ampere,  the  unit 
of  electric  current  which  has  the  value  one-tenth  (0.1)  in  terms 
of  the  centimeter,  gram  and  second;  (3)  the  Volt,  the  unit  of 
electromotive  force  which  has  the  value  100,000,000  in  terms 
of  the  centimeter,  the  gram,  and  the  second;  (4)  the  Watt,  the 
Unit  of  Power,  which  has  the  value  10,000,000  in  terms  of  the 
centimeter,  the  gram,  and  the  second. 

II.  As  a  system  of  units  representing  the  above  and  suffi- 
ciently near  to  them  to  be  adopted  for  the  purpose  of  elec- 
trical measurements  and  as  a  basis  for  legislation,  the  Con- 
ference recommends  the  adoption  of  the  International  Ohm, 
the  International  Ampere,  and  the  International  Volt,  de- 
fined according  to  the  following  definitions. 

III.  The  Ohm  is  the  first  Primary  Unit. 

IV.  The  International  Ohm  is  defined  as  the  resistance  of 
a  specified  column  of  mercury. 

V.  The  International  Ohm  is  the  resistance  offered  to  an 
unvarying  electric  current  by  a  column  of  mercury  at  the 
temperature  of  melting  ice,  14.4521  grams  in  mass,  of  a  con- 
stant cross-sectional  area  and  of  a  length  of  106.300  cm. 

To  determine  the  resistance  of  a  column  of  mercury  in  terms 
of  the  International  Ohm,  the  procedure  to  be  followed  shall 
be  that  set  out  in  specification  I,  attached1  to  these  resolutions. 

VI.  The  Ampere  is  the  second  Primary  Unit. 

VII.  The  International  Ampere  is  the  unvarying  electric 
current  which,  when  passed  through  a  solution  of  nitrate  of 
silver  in  water,  in  accordance  with  the  specification  II,  at- 
tached1 to  these  resolutions,  deposits  silver  at  the  rate  of 
0.00111800  of  a  gram  per  second. 

VIII.  The   International  'Volt   is   the   electrical    pressure 
*See  Elec.  Review,  vol.  63.  p.  738. 


INTRODUCTION  9 

which,  when  steadily  applied  to  a  conductor  whose  resistance 
is  one  International  Ohm,  will  produce  a  current  of  one  Inter- 
national Ampere. 

IX.  The  International  Watt  is  the  energy  expended  per 
second  by  an  unvarying  electric  current  of  one  International 
Ampere  under  an  electric  pressure  of  one  International  Volt. 

The  Conference  recommends  the  use  of  the  Weston  Normal 
Cell  as  a  convenient  method  of  measuring  both  electromotive 
force  and  current,  and  when  set  up  under  the  conditions  speci- 
fied in  schedule  C  may  be  taken,  provisionally,  as  having,  at 
a  temperature  of  20°  C.,  an  E.  M.  F.  of  1.0184  volts.  (1.0183 
since  Jan.  1,  1911). 

In  cases  in  which  it  is  not  desired  to  set  up  the  Standards 
provided  in  the  resolutions  above,  the  Conference  recommends 
the  following  as  working  methods  for  the  realization  of  the 
International  Ohm,  the  Ampere  and  the  Volt. 

1.  For  the  International  Ohm. 

The  use  of  copies,  constructed  of  suitable  material  and  of 
suitable  form  and  verified  from  time  to  time,  of  the  Inter- 
national Ohm,  its  multiples  and  submultiples. 

2.  For  the  International  Ampere. 

(a)  The  measurement  of  current  by  the  aid  of  a  current 
balance  standardized  by  comparison  with  a  silver  voltameter. 

(6)  The  use  of  a  Weston  Normal  Cell  whose  electromotive 
force  has  been  determined  in  terms  of  the  International  Ohm 
and  International  Ampere,  and  of  a  resistance  of  known  value 
in  International  Ohms. 

3.  For  the  International  Volt. 

(a)  A  comparison  with  the  difference  of  electrical  potential 
between  the  ends  of  a  coil  of  resistance  of  known  value  in 
International  Ohms,  when  carrying  a  current  of  known  value 
in  International  Amperes. 

(6)  The  use  of  a  Weston  Normal  Cell  whose  electromotive 
force  has  been  determined  in  terms  of  the  International  Ohm 
and  the  International  Ampere. 


CHAPTER  I 
AMMETER  AND  VOLTMETER  METHODS 

12.  Laws  of  Electric  Currents — Use  of  an  Ammeter. — The 
Weston  ammeter  is  a  good  and  accurate  instrument  for  the 
measurement  of  electric  current.  It  is  a  very  delicate  and  sen- 
sitive instrument  and  must  always  be  handled  with  care. 
Mechanical  shocks  or  jars  will  injure  the  jeweled  bearings, 
and  too  large  a  current  through  it  will  wrench  the  movable 
coil  and  bend  the  delicate  pointer,  even  if  the  instrument  is 
not  burned  out  thereby. 

When  it  is  desired  to  use  the  ammeter  for  the  measurement 
of  current  it  is  connected  in  series  with  the  rest  of  the  circuit, 
and  therefore  the  entire  current  passes  through  the  instrument. 
Great  care  should  always  be  exercised  never  to  allow  a  larger 
current  to  flow  through  an  ammeter  than  it  is  intended  to 
carry.  It  is  always  best  to  have  a  key  in  the  circuit  and  while 
keeping  the  eye  on  the  needle  of  the  ammeter  tap  the  key 
gently,  thus  closing  it  for  a  fraction  of  a  second  only.  If  the 
needle  does  not  move  very  far  the  key  can  be  held  down  for  a 
longer  time,  and  if  it  is  now  seen  that  the  needle  will  remain 
on  the  scale  the  key  can  be  held  down  until  the  needle  comes 
to  rest.  Back  of  the  needle  is  a  strip  of  mirror,  and  by  placing 
the  eye  in  such  a  position  that  the  image  of  the  needle  is  hidden 
by  the  needle  itself,  the  error  due  to  parallax  in  reading  the 
scale  can  be  avoided. 

The  scales  of  these  instruments  are  graduated  to  read  the 
current  directly  in  amperes.  Sometimes  the  pointer  does  not 

10 


AMMETER  AND  VOLTMETER  METHODS 


11 


stand  at  the  zero  of  the  scale  when  no  curent  is  flowing.  When 
this  is  the  case  the  position  of  rest  should  be  carefully  noted 
and  the  observed  reading  corrected  accordingly. 

For  this  exercise  join  a  dry  cell,  a  coil  of  several  ohms  re- 
sistance, a  key,  and  the  ammeter  in  series,  that  is,  one  after 
the  other  to  form  a  single  and  continuous  circuit.  The  current 


F  +, 


E 


FIG.  1. — Resistances  in  series. 

flows  from  the  positive,  or  carbon,  pole  of  the  cell  and  should 
enter  the  ammeter  at  the  post  marked  + .  Measure  and  record 
the  value  of  the  current  at  different  points  along  this  circuit, 
to  determine  whether  the  current  has  the  same  value  through- 
out its  path  or  whether  it  is  smaller  after  passing  through  the 
resistances.  Next  add  the  remaining  coils  to  the  circuit, 


Ba 


FIG.  2. — Resistances  R3  and  Rz  in  parallel. 

keeping  them  all  in  series,  and  note  the  value  of  the  current 
at  the  same  points  as  before.  State  in  your  own  words  the 
effect  of  adding  resistance  to  the  circuit. 

Remove  one  of  the  coils  and  connect  it  in  parallel  with  one 
of  those  still  remaining  in  the  circuit,  i.e.,  so  that  the  current  in 
the  main  circuit  will  divide,  a  part  going  through  each  of  the 


12  ELECTRICAL  MEASUREMENTS 

two  coils  in  parallel.  Measure  the  part  of  the  current  through 
each  coil:  also  the  main  current.  Could  you  have  foretold 
the  value  of  the  latter  without  measuring  it? 

This  exercise  should  show  that  the  only  thing  in  common  to 
coils  in  series  is  the  value  of  the  current  passing  through  them. 
For  this  reason  an  ammeter  is  always  joined  in  series  with  the 
circuit  in  which  the  value  of  the  current  is  desired.  Record 
the  data  in  a  neat  tabular  form  similar  to  the  following : 

To  MEASURE  THE  CURRENT  ALONG  AN  ELECTRIC  CIRCUIT 


Ammeter 
zero 

Ammeter 
reading 

Current 

Point  at  which 
current  is  measured 

13.  Fall  of  Potential  in  Electric  Circuit— Use  of  a  Voltmeter.— 
The  Weston  voltmeter  is  a  good  and  accurate  instrument  for 
the  measurement  of  electromotive  forces.  It  is  a  very  delicate 
and  sensitive  instrument  and  must  always  be  handled  with 
care.  Mechanical  shocks  or  jars  will  injure  the  jeweled  bear- 
ings, and  too  large  a  current  through  it  will  wrench  the  movable 
coil  and  bend  the  delicate  pointer,  even  if  the  E.M.F.  is 
not  much  larger  than  that  intended  to  be  measured  by  the 
instrument. 

Since  a  voltmeter  is  always  used  to  measure  the  difference  of 
potential  between  two  points  it  is  not  put  into  the  circuit  like  an 
ammeter,  but  the  two  binding  posts  of  the  voltmeter  are  con- 
nected directly  to  the  two  points  whose  difference  of  potential 
is  desired.  The  voltmeter  thus  forms  a  shunt  circuit  between 
the  two  points.  There  is  a  current  through  the  voltmeter 
proportional  to  the  difference  of  potential  between  the  two 
points  to  which  it  is  joined.  This  current  passing  through 
the  movable  coil  of  the  instrument  deflects  the  pointer  over 


AMMETER  AND  VOLTMETER  METHODS 


13 


the  scale,  but  the  latter  is  graduated  to  read,  not  the  current, 
but  the  number  of  volts  between  the  two  binding  posts  of  the 
voltmeter.  In  some  instruments  there  is  a  strip  of  mirror 
placed  below  the  needle.  When  the  eye  is  so  placed  that  the 
image  of  the  needle  is  hidden  by  the  needle  itself,  the  reading 
can  be  taken  without  the  error  due  to  parallax.  Sometimes 
the  pointer  does  not  stand  at  the  zero  of  the  scale  when  there 
is  no  current  through  the  instrument.  When  this  is  the  case 
the  position  of  rest  should  be  carefully  noted  and  the  observed 
reading  corrected  accordingly. 

For  this  exercise  join  a  cell,  two  coils  of  several  ohms  resis- 
tance, and  a  key,  in  series.  With  the  voltmeter  measure  the 
fall  of  potential  over  each  coil, 
also  over  both  together.  Add  a 
third  coil  and  measure  the  fall  of 
potential  over  each ;  also  over  all. 
Note  where  these  last  readings 
are  the  same  as  before  and  where 
they  are  different.  Add  a  sec- 
ond cell  and  repeat  the  above 
readings.  Note  changes. 

Join  two  of  the  coils  in  parallel, 
thus  forming  a   divided   circuit 
and  allowing  a  part  of  the  cur- 
rent to  flow  through  each  branch, 
tial  over  each  branch. 


FIG.  3. — Connections  for  a 
voltmeter. 


Measure  the  fall  of  poten- 
Add  a  third  coil  in  parallel  with  the 
other  two  and  again  measure  the  fall  of  potential  over  each. 

This  exercise  should  show,  especially  in  connection  with 
the  preceding  one,  that  the  only  thing  in  common  to  several 
circuits  in  parallel  is  that  each  one  has  the  same  fall  of  potential. 
Hence  a  voltmeter  is  always  joined  in  parallel  with  the  coil,  the 
fall  of  potential  over  which  is  desired.  For  the  voltmeter 
indicates  the  fall  of  potential  over  itself;  and  if  it  forms  one 
of  the  parallel  circuits,  the  fall  of  potential  over  each  one  is 
the  same  as  that  indicated  by  the  voltmeter. 
Record  the  data  as  below: 


14 


ELECTRICAL  MEASUREMENTS 
FALL  OF  POTENTIAL  IN  AN  ELECTRIC  CIRCUIT 


Voltmeter 

Fall  of 
potential 

Position  of 
voltmeter 

Zero 

Reading 

14.  Ohm's  Law.  —  The  current  that  flows  through  any  con- 
ductor is  found  experimentally  to  be  directly  proportional  to 
the  potential  difference  between  its  terminals.  This  state- 
ment was  first  formulated  in  1827  by  Dr.  Ohm,  and  is  known 
as  Ohm's  Law.  It  is  usually  written, 


where  V  denotes  the  potential  difference  over  the  circuit 
through  which  is  flowing  the  current  I.  The  factor  R  is  called 
the  resistance  of  the  conductor,  and  its  value  depends  only  upon 
the  dimensions  and  material  of  the  wire  and  its  temperature. 
It  is  entirely  independent  of  V  and  7. 

This  relation  holds  equally  well  whether  the  entire  circuit 
is  considered  or  whether  only  a  portion  of  such  circuit  is  taken. 
In  the  former  case  the  law  states  that  the  current  through  the 
circuit  is  equal  to  the*  total  E.M.F.  in  the  circuit  divided  by 
the  resistance  of  the  entire  circuit,  including  that  of  the  battery 
and  the  connecting  wires.  When  applied  to  a  single  conductor, 
AB,  the  law  states  that  the  current  flowing  through  the  con- 
ductor is  equal  to  the  fall  of  potential  between  A  and  B  divided 
by  the  resistance  AB. 

To  determine  the  resistance  of  a  conductor  it  is  then  only 
necessary  to  measure  with  an  ammeter  the  current  flowing 
through  it,  and  with  a  voltmeter  measure  the  difference  of 
potential  between  its  terminals.  In  case  the  current  is  at  all 


AMMETER  AND  VOLTMETER  METHODS 


15 


variable  the  two  instruments  must  be  read  at  the  same  time, 
for  Ohm's  law  applies  only  to  simultaneous  values  of  the  current 
and  voltage. 

15.  Measurement  of  Resistance  by  Ammeter  and  Volt- 
meter. First  Method. — Join  the  conductor  whose  resistance, 
R,  is  to  be  measured,  in  series  with  an  ammeter,  Am,  a,  key, 
a  battery,  and  sufficient  auxiliary  resistance  to  keep  the 
current  from  being  too  large.  The  current  should  enter  the 
ammeter  at  the  post  marked  +.  Then  keeping  the  eye  fixed 
on  the  needle  of  the  ammeter,  close  the  key  for  a  fraction  of  a 
second.  If  the  deflection  is  in 
the  right  direction,  and  is  not 
too  large,  the  key  can  be  closed 
again  and  the  value  of  the  cur- 
rent read  from  the  scale  of  the  am- 
meter. Should  the  current  be 
too  large,  the  auxiliary  resistance 
can  be  increased  until  the  cur- 
rent is  reduced  to  the  desired 
value. 

To  measure  the  fall  of  poten- 
tial over  the  conductor  its  two 


FIG.  4. — To  measure  a  small 
resistance,  R. 


terminals  are  joined  to  the  binding  posts  of  the  voltmeter,  by 
means  of  the  additional  wires.  That  terminal  of  the  resistance 
at  which  the  current  enters  should  be  joined  to  the  voltmeter 
post  marked  +.  Close  the  key  for  an  instant  as  before,  keep- 
ing the  eye  on  the  voltmeter  needle,  and  if  the  deflection  is  in 
the  right  direction  and  not  too  large  the  reading  of  the  volt- 
meter can  be  taken. 

Now  close  the  key  and  record  simultaneous  readings  of  the 
ammeter  and  the  voltmeter.  Do  this  three  times,  changing  the 
current  slightly  by  means  of  the  auxiliary  resistance  before 
each  set  of  readings.  Compute  the  resistance  of  R  from  each 
set  of  readings  by  means  of  the  formula 

V 
I 


R=  4 


16 


ELECTRICAL  MEASUREMENTS 


where  V  and  I  are  the  voltmeter  and  ammeter  readings,  cor- 
rected for  the  zero  readings.  The  mean  of  these  three  results 
will  be  the  approximate  value  of  the  resistance. 

A  more  exact  value  of  R  can  be  obtained  by  correcting  the 
current  as  measured  by  the  ammeter  for  the  small  current,  i, 
which  flows  through  the  voltmeter  .  The  current  through  R 
is,  strictly,  not  7,  but  I— i.  This  gives  then. 


R  = 


I-i 


16.  Measurement  of  Resistance  by  Ammeter  and  Volt- 
meter. Second  Method.  This  method  differs  from  the  First 
Method  by  the  position  of  the  voltmeter.  In  the  first  method 


FIG.  5. — To  measure  a  large  resistance,  R. 

the  ammeter  measured  both  the  current  through  R  and  the 
small  current  through  the  voltmeter,  and  therefore  its  readings 
were  somewhat  too  high. 

If  the  connections  are  made  as  shown  in  the  figure  this  error 
is  avoided,  as  the  current  now  passing  through  R  is  strictly 
the  same  as  that  measured  by  the  ammeter.  The  voltmeter, 
however,  now  measures  the  fall  of  potential  over  both  R  and 


AMMETER  AND  VOLTMETER  METHODS 


17 


the  ammeter,  and  therefore  the  resistances  of  both  are  meas- 
ured together.  The  resistance  of  R  is  then  found  by  subtract- 
ing the  resistance  of  the  ammeter  from  the  measured  amount. 
The  formula  then  becomes, 


where  A  is  the  resistance  of  the  ammeter. 

Measure  the  resistance  of  two  coils  and  check  results  by 
also  measuring  them  in  series  and  in  parallel.  The  measured 
resistances  should  be  compared  with  the  values  computed 
from  the  formula,  for  series,  R  =  Ri  +  R%  and  for  parallel 


Ri  +  Rz 
Record  the  data  aa  follows: 

To  MEASURE  THE  RESISTANCE  OF  . 


Resistance 

Coil 

Ammeter 

Voltmeter 

y- 

measured 

reading 

reading 

v 

7 

Corrected 

17.  To  Find  the  Best  Arrangement  for  Measuring  Resist- 
ance with  an  Ammeter  and  a  Voltmeter. — In  each  of  the  pre- 
ceding methods  for  measuring  resistance  by  means  of  an  am- 
meter and  a  voltmeter  it  is  necessary  to  apply  a  correction 
to  the  observed  readings  in  order  to  obtain  the  true  value  of 
the  resistance  being  measured.  It  is  the  object  of  this  sec- 
tion to  inquire  under  what  conditions  these  corrections  are  a  min- 
imum. In  order  to  compare  the  two  correction  terms  with  each 
other  they  will  both  be  expressed  in  the  form  of  factors  by 


18  ELECTRICAL  MEASUREMENTS 

which  the  observed  values  are  multiplied  to  obtain  the  true 
values. 

In  the  first  method  the  correction  is  applied  to  the  ammeter 
reading,  the  true  current  through  R  being  /  —  i.  The  true 
value  of  the  resistance  is,  then, 

P     v       v          v        R> 

1      1       i~~     /         A  ~      /          P'\  "  P' 

'III  \      1  -  — 

l  ~~ 


where  S  is  the  resistance  of  the  voltmeter,  and  R'  is  the  uncor- 
rected  value  of  R. 

In  the  second  method  the  value  of  R  ig 

A  \ 

(2) 

where  A  is  the  resistance  of  the  ammeter,  and  Rr,  as  before, 
denotes  the  ratio  of  V  to  7. 

The  effect  of  the  correction  term  in  (1)  is  to  add  something 
to  the  measured  value  of  the  resistance,  while  the  correction 
term  in  (2)  reduces  it.  In  using  an  ammeter  and  a  voltmeter 
to  measure  resistance,  that  arrangement  should  be  chosen, 
therefore,  which  involves  the  smaller  correction  factor.  Com- 
paring these  terms  it  is  seen  that  the  former  is  small,  that  is, 
it  is  near  unity,  when  R'  is  small;  and  the  second  has  little 
effect  when  R'  is  large. 

The  effect  of  these  corrections  will  be  the  same  when  they 
are  equal,  that  is,  when 

!-£-==  1-^7  (3) 

and  this  is  the  case  when 

R'  =   VZS. 

For  resistances  smaller  than  this  the  first  method  has  the 
smaller  correction.  For  larger  resistances  the  second  method 
should  be  used.  Whether  the  correction  is  applied  or  not,  it 
should  always  be  made  as  small  as  possible.  If  the  proper 
method  is  selected  the  correction  will  almost  never  be  as  large 
as  1  per  cent. 


AMMETER  AND  VOLTMETER  METHODS 


19 


18.  Internal  Resistance  of  a  Battery.  —  The  internal  resist- 
ance of  a  battery  is  readily  measured  by  the  ammeter  and 
voltmeter  method.  The  cell  is  joined  in  series  with  a  suitable 
resistance,  a  key,  and  an  ammeter,  as  shown  in  Fig.  6.  A  volt- 
meter is  joined  in  parallel  with  the  resistance  and  key.  When 
the  key  is  closed  the  current  drawn  from  the  cell  is  by  Ohm's  law. 


T 


R  +  r 
from  which  it  follows  that 

E-IR      E-E' 


,.. 

(A) 


Ba 
.  + 


FIG.  6.  —  To  measure  the  internal  resistance  of  the  cell,  Ba. 

E'  (=  IR)  being  the  reading  of  the  voltmeter  when  the  key 
is  closed.  The  current  is  measured  by  the  ammeter.  When 
the  key  is  opened  the  voltmeter  is  connected  directly  to  the 
battery  and  therefore  measures  E. 

If  the  value  of  R  is  known  the  ammeter  can  be  omitted, 
and  the  value  of  the  „  current  determined  from  the  relation 
/  =  E'  /R.  Equation  (A)  then  becomes 


(B) 


20  ELECTRICAL  MEASUREMENTS 

As  there  is  apt  to  be  some  polarization  of  the  cell  it  is  best  to 
close  the  key  only  long  enough  to  obtain  simultaneous  readings 
of  the  ammeter  and  voltmeter.  The  value  of  E  is  then  read 
immediately  upon  opening  the  key. 

19.  A  More  Exact  Method.  —  Inasmuch  as  there  is  a  small 
current  through  the  cell  and  voltmeter  when  the  key  is  open, 
the  foregoing  method  is  subject  to  a  corresponding  correction. 
The  value  of  this  current  is 


~  S  +  r 

where  S  denotes  the  resistance  of  the  voltmeter  and  r  includes 
not  only  the  resistance  of  the  battery  but  also  that  of  the 
ammeter.  This  may  be  written, 

E  =  Si  +  ri. 

The  voltmeter  measures  the  part  Si,  which  is  the  fall  of  poten- 
tial over  its  own  resistance.  The  part  ri  is  the  fall  of  potential 
within  the  cell  and  will  be  negligible  only  when  r  is  very  small. 
If  V  is  the  voltmeter  reading, 

V  =  Si  =  E  -  ri. 

Similarly  when  the  key  is  closed  and  a  larger  current  taken  from 
the  cell,  we  have  from  (A) 

E'  =  E  -  ri. 
From  these, 

r  =  y-E'  (C) 

I-i 

where  i  is  the  voltmeter  current  with  the  key  open.  This 
reads, 

Change  in  Potential  Difference 
Internal  Resistance  =  --  7^7—    —  -  —  ?s  — 

Change  in  Current 

which  is  a  simple  deduction  from  Ohm's  law.  E'  and  I  are 
simultaneous  readings  of  the  voltmeter  and  the  ammeter  with 
the  key  closed.  V  and  i  are  the  corresponding  readings  with 


AMMETER  AND  VOLTMETER  METHODS 


21 


the  key  open.  Thus  the  zero  readings  of  both  instruments  are 
eliminated.  Since  the  resistance  of  the  ammeter  was  included 
in  r,  it  must  now  be  subtracted  from  the  computed  result  to 
obtain  the  resistance  of  the  cell  alone.  Thus 


r'  =  r  -  A. 


Record  data  as  follows : 


INTERNAL  RESISTANCE  OF CELL 

Am.  zero  =  Vm.  zero  =  Am.  resistance  = 


Kind  of  cell 

R 

Key  closed 

Key  open 

r 

r' 

E' 

I 

V 

i 

20.  Relation  Between  Available   E.M.F.   and   Current. — 

Ohm's  law  when  applied  to  a  complete  circuit  gives  the  rela- 
tion, 

E 


or, 


E  =  RI  +  Ir  =  E'  +  Jr. 


(A) 


The  term  Ir  is  the  fall  of  potential  over  the  internal  resist- 
ance of  the  cell.  RI  is  the  fall  of  potential  over  the  entire 
external  part  of  the  circuit,  and  is  often  denoted  by  the  single 
symbol  E'.  Various  names  have  been  applied  to  this  term 
E',  such  as  terminal  E.M.F.,  terminal  potential  difference, 
pole  potential,  available  E.M.F.,  etc.  It  is  that  part  of  the 
total  E.M.F.  of  the  cell  that  is  available  for  doing  useful  work. 
Its  value,  from  eq.  A  is, 

W  =  E  -  Ir, 


22 


ELECTRICAL  MEASUREMENTS 


and  from  this  it  appears  that  the  available  E.M.F.  is  less  as 
the  current  becomes  larger. 

A  battery  is  joined  to  a  resistance  box  and  ammeter  in 
series.  By  using  various  values  of  R  the  current  can  be  varied 
throughout  its  possible  range,  the  values  being  read  from  the 
ammeter.  A  voltmeter  joined  to  the  poles  of  the  battery  gives 
the  corresponding  values  of  Ef.  The  key  should  be  kept 
closed  as  little  as  possible  to  avoid  unnecessary  polarization 
of  the  battery.  The  following  values  of  R  will  give  a  good 


FIG.  7. — To  measure  the  available  E.M.F.  of  the  battery  Ba  correspond- 
ing to  the  current  through  it. 


series  of  readings:  100,  60,  40,  30,  20,  16,  13,  10,  8,  7,  5,  4,  3, 
2,  1,  ohms. 

From  the  values  of  E'  and  I  plot  a  curve.  Repeat  for  one 
or  two  different  types  of  cells  using  at  least  one  cell  of  low 
internal  resistance.  Determine  from  the  curves  the  maxi- 
mum current  which  each  cell  can  furnish.  Find  one  reason 
why  an  ammeter  should  not  be  joined  to  the  poles  of  a  cell 
as  is  done  with  a  voltmeter. 
Record  the  data  as  follows: 


AMMETER  AND  VOLTMETER  METHODS  23 

RELATION  BETWEEN  AVAILABLE  E.M.F.  AND  CURRENT  FOR  A.  .    .  .CELL 


R 

Ammeter 

I 

Voltmeter 

E' 

Zero 

Reading 

Zero 

Reading 

21.  Useful  Power  from  a  Cell. — Electrical  power  is  measured 
by  the  product  El,  where  /  denotes  the  current  and  E  the  fall 


FIG.  8. — Measurement  of  power. 

of  potential  over  the  circuit  in  which  the  power  is  being  ex- 
pended. The  unit  in  which  power  is  measured  is  the  watt, 
one  watt  being  the  product  of  one  volt  by  one  ampere. 

When  a  battery  is  furnishing  a  current  the  total  power 
expended  is  supplied  by  the  chemical  reactions  within  the  cell. 
Part  of  this  power  is  expended  in  the  external  circuit  where  it 
may  be  used  in  running  motors  or  doing  other  useful  work. 
The  remainder  is  spent  within  the  cell  and  only  goes  to  warm- 


24 


ELECTRICAL  MEASUREMENTS 


ing  the  contents  of  the  battery.     In  some  cases  the  greater 
part  of  the  energy  is  thus  wasted  within  the  cell. 

The  object  of  this  experiment  is  to  measure  the  power  in  the 
external  circuit  when  various  currents  are  flowing.  The  cell 
is  joined  in  series  with  an  ammeter  and  a  resistance  which 
will  carry  the  largest  current  that  may  be  used.  A  voltmeter 
measures  the  fall  of  potential.  Probably  there  will  be  found 
a  point  beyond  which  the  useful  power  decreases  even  through 


100%- 


FIQ.  9. — Power  from  a  battery. 

the  current  is  made  larger.  Since  the  current  is  proportional 
to  the  amount  of  chemicals  used  up  in  the  cell,  it  will  be  well  to 
express  the  results  as  a  function  of  the  current.  This  is  best 
done  by  means  of  a  curve,  using  values  of  the  current  for 
abscissae  and  the  corresponding  values  of  power  for  the 
ordinates. 

The  equation  of  this  curve  is 


W  =  IE'  =  IE  -  Pr. 


AMMETER  AND  VOLTMETER  METHODS 


25 


Placing  the  derivative  of  W  with  respect  to  /  equal  to  zero 
gives 

dW 


dl 


=  E  -  2Ir  =  0. 


as  the  condition  for  obtaining  the  maximum  power.  This  is 
when  /  =  E/2r,  or  the  maximum  available  power  is  obtained 
when  the  external  resistance  is  equal  to  the  internal  resistance 
of  the  battery. 

Plot  also  the  curve  showing  the  total  power,  El,  supplied 
by  the  cell.  The  ratio  of  the  ordinates  of  these  two  curves, 
that  is,  the  ratio  of  the  useful  power  to  the  total  power,  gives 
the  efficiency  of  the  cell. 

The  efficiency  can  also  be  plotted  as  a  curve  on  the  same 
sheet  with  the  other  two  curves. 
Record  the  data  as  follows: 

USEFUL  POWER  FROM  A  .  .  CELL 


R 

Ammeter 

7 

Voltmeter 

E' 

W 

Zero 

Reading 

Zero 

Reading 

Problem. — How  much  power  is  expended  in  an  ammeter  of  0.01 
ohm  resistance  if  it  is  connected  to  a  storage  battery  of  ten  volts  and 
0.01  ohm  internal  resistance?  Why  is  it  best  not  to  try  the  actual  ex- 
periment in  the  laboratory? 

22.  E.M.F.  of  a  Cell  by  a  Voltmeter  and  an  Auxiliary 

Battery. — When  a  cell,  B,  is  joined  to  a  voltmeter  the  reading 
of  the  latter  does  not  strictly  give  the  E.M.F.  of  the  cell. 
This  discrepancy  is  small  for  cells  of  low  internal  resistance, 
but  in  many  cases  it  can  not  be  neglected.  If  the  resistance  of 


26 


ELECTRICAL  MEASUREMENTS 


the  voltmeter  is  S,  and  that  of  the  cell  is  r,  then  by  Ohm's 
law  the  small  current  through  the  cell  and  voltmeter  is, 

E 


t  = 


From  this  the  E.M.F.  of  the  cell  is, 
E  =  Si  +  ri, 

where  Si  is  the  fall  of  potential  over  the  external  circuit,  in 
this  case  the  voltmeter.    Since  a  voltmeter  measures  only  the 
fall  of  potential  over  its  own  resistance  the  reading  of  the 
voltmeter  is  not  E,  but  Si,  which  is  less 
than  E  by  the  amount  ri.    When  r  is 
small  this  term  can  be  neglected,  but  not 
otherwise,  unless  i  can  be  made  very  small. 
In  the  following  method  no  current  is 
taken  from  the  cell  and  therefore  its  full 
E.M.F.  can  be  measured.    A  second  cir- 
cuit is  formed,  consisting  of  the  voltmeter, 
two  cells,  and  a  resistance  R,  which  is  ad- 
justed until  the  voltmeter  reading  with  K 

FIG.  lO.-To  measure  °Pen  is  about  e(lual  to  the  E'M'R  °f  B' 
the  E.M.F.  of  B.       There  is  thus  introduced  into  the  former 

circuit  of  B  an  additional  difference  of 
potential,  viz.^  SI,  the  fall  of  potential  over  the  voltmeter 
due  to  the  current  I  of  the  second  circuit.  When  K  is  closed 
the  current  through  the  first  circuit  will  now  be, 

E  -SI 


This  current  i  flowing  through  the  voltmeter  in  addition  to 
the  current  I,  will  cause  an  increased  deflection.  It  is  possible 
to  adjust  R  to  such  a  value  that  closing  K  will  not  affect  the 
reading  of  the  voltmeter;  in  other  words  it  is  possible  to  make 
i  =  0.  This  means  that 


AMMETER  AND  VOLTMETER  METHODS 


27 


and  SI,  being  the  fall  of  potential  over  the  voltmeter  is  given 
directly  by  the  voltmeter  reading. 

Measure  in  this  way  the  E.M.F.  of  several  cells  and  com- 
pare the  results  obtained  in  each  case  with  the  readings  of 
the  voltmeter  when  used  alone. 
The  data  may  be  recorded  as  follows: 


Name  of  cell 

Voltmeter  readings 

R 

Alone 

With  aux.  battery 

AAAA 


23.  Measurement  Of  Current  by  a  Voltmeter  and  Shunt. — 
When  a  current  I  flows  through  a  resistance  R}  the  fall  of 
potential  over  R  is  E  =  IR,  which 
is  in  accordance  with  Ohm's  law 
and  our  definition  of  the  term 
fall  of  potential.  In  Article  15 
both  E  and  /  were  measured 
and  the  value  of  R  was  then  com- 
puted. When  R  is  known  the 
experiment  can  be  reversed  and 
by  measuring  E  the  value  of  / 
can  be  computed.  Thus  a  volt- 
meter may  be  used  to  measure  currents  in  place  of  an  am- 
meter. The  arrangement  is  shown  in  Fig.  11. 

Usually  the  resistance  which  would  be  suitable  for  this 
purpose  would  be  an  ohm  or  less.  Therefore  any  variation  in 
the  position  of  the  voltmeter  connections  at  a  and  b  would 
cause  considerable  change  in  the  resistance  between  these 
points.  It  is  best  to  have  these  connections  soldered  fast,  and 
let  the  voltmeter  connections  be  made  at  the  auxiliary  points 


FIG.  11. — Measurement  of  cur- 
rent. 


28 


ELECTRICAL  MEASUREMENTS 


a'  and  6'  where  a  little  resistance,  more  or  less,  in  the  volt- 
meter circuit  will  be  inappreciable. 

Such  shunts  are  often  made  having  a  resistance  of  0.1, 0.01, 
0.001  or  less,  of  an  ohm.  The  current  is  then  10,  100,  1000,  or 
more,  times  the  voltmeter  reading.  This  principle  is  also  used 
in  the  construction  of  ammeters  for  measuring  large  currents. 
The  greater  part  of  the  current  is  carried  by  a  shunt  of  low 
resistance,  while  the  delicate  moving  coil  carries  only  a  small 
current  and  thus  in  reality  acts  as  a  sensitive  voltmeter.  The 
numbers  on  the  scale,  however,  instead  of  reading  volts,  are 
made  to  give  the  corresponding  values  of  the  currents  passing 
through  the  instrument. 

24.  Measurement  of  a  High  Resistance  by  a  Voltmeter 
Alone. — This  method  is  a  modification  of  the  ammeter  and 
voltmeter  method  for  the  measurement  of  moderate  resist- 
ances, and  it  is  based  on  the  fact 
that   a  voltmeter  is  really  a  very 
sensitive  ammeter.     It  can  be  used 
as  an  ammeter  whenever  the  high 
resistance  in  series  with  the  moving 
coil  will  be  only  a  small  part  of  the 
total  resistance  in  the  circuit. 

The  voltmeter  is  joined  in  series 
with  a  battery  and  the  high  resis- 
tance to  be  measured.  In  this  way 

it  serves  as  an  ammeter  to  measure  the  current  through  the 
circuit.  The  reading,  V,  of  the  voltmeter  gives  the  fall  of  po- 
tential over  its  own  resistance  S,  or  in  symbols, 

V  =  SI. 


FIG.    12. — To    measure   the 
high  resistance,  H.  R. 


Considering  the  entire  circuit,  the  value  of  the  current  is 
given  by  the  expression 


E 


8- 


AMMETER  AND  VOLTMETER  METHODS  29 

Eliminating  /  from  these  two  equations  and  solving  for  R  gives, 

E  -  V 


The  value  of  E,  the  E.M.F.  of  the  battery,  is  easily  measured 
by  the  same  voltmeter  by  connecting  it  directly  to  the  battery. 
A  key  arranged  to  short  circuit  the  high  resistance  will  readily 
change  the  voltmeter  from  its  position  as  an  ammeter  to  that 
of  a  voltmeter. 

Record  data  as  follows: 

To  MEASURE  THE  RESISTANCE  OF  ........ 

Vm.  Zero  =  Vm.  Resist.  =        ohms 


Name  of 

object  measured 

V 

E 

R 

25.  Time  Test  of  a  Celt— The  time  test  6"f  a  cell  is  designed 
to  show  how  well  the  cell  can  maintain  a  current,  how  effec- 
tive is  the  action  of  the  depolarizer,  and  the  rapidity  and 
extent  of  the  recovery  of  the  cell  when  the  current  ceases. 
Such  a  test  usually  continues  for  an  hour,  and  the  results  may 
best  be  shown  graphically  by  means  of  curves.  These  curves 
should  show  the  value  of  the  E.M.F.  of  the  cell  at  each  in- 
stant during  the  test;  the  available  E.M.F.;  the  current;  and 
the  internal  resistance  of  the  cell.  After  maintaining  the  cur- 
rent for  an  hour,  the  circuit  is  opened  and  the  cell  allowed  to 
recover.  The  curves  should  show  the  rate  and  extent  of  this 
recovery. 

Of  course  it  is  evident  that  such  a  continued  test  may  prove 
rather  severe  for  cells  intended  for  only  a  few  minutes'  use  at 
one  time.  And  on  the  other  hand,  cells  which  make  a  good 
showing  under  the  test  might  not  be  the  best  for  long  continued 
service.  Nevertheless  such  a  test  gives  the  most  informa- 


30  ELECTRICAL  MEASUREMENTS 

tion  regarding  the  behavior  of  the  cell  that  can  be  obtained 
in  the  short  space  of  2  hours. 

The  set  up  for  making  a  time  test  is  the  same  as  that  for 
measuring  the  internal  resistance  of  the  cell,  save  that  the 
battery  circuit  remains  closed  all  of  the  time  except  when  it  is 
opened  for  an  instant  to  measure  the  E.M.F.  of  the  cell.  With 
the  proper  preparation  beforehand  it  is  not  difficult  to  observe 
all  the  necessary  data  and  record  it  in  a  neat  and  convenient 
form. 

Readings  should  be  taken  once  a  minute.  Practice  in  doing 
this  should  be  obtained  by  measuring  the  E.M.F.  of  the  cell 
before  any  current  is  drawn  from  it.  When  ready  to  commence 
the  test  proper,  the  circuit  is  closed  through  5  or  10  ohms,  or 
about  as  much  resistance  as  the  internal  resistance  of  the  cell, 
and  as  soon  thereafter  as  possible  the  first  reading  for  the  avail- 
able E.M.F.  is  taken.  One  minute  later  the  circuit  is  opened 
just  long  enough  to  take  a  reading  of  the  E.M.F.  of  the  cell. 
Thus  every  2  minutes  the  available  E.M.F.  is  recorded  and 
on  the  intermediate  minutes  the  value  of  the  total  E.M.F. 
is  measured. 

From  this  data  two  curves  are  plotted — one  showing  the 
variation  in  the  E.M.F.  of  the  cell  during  the  test,  and  the 
other  showing  the  same  with  respect  to  the  available  E.M.F. 
The  internal  resistance  is  computed  at  5-minute  intervals, 
the  values  of  E  and  E'  being  taken  from  the  curves.  The  cur- 
rent is  computed  from  the  values  of  R  and  the  available 
E.M.F. 

After  the  first  hour  of  the  test  the  battery  key  is  changed 
so  as  to  keep  the  circuit  open,  except  for  an  instant  each 
minute  when  it  is  closed  long  enough  to  read  the  voltmeter. 
The  keys  can  be  worked  by  hand  and  if  proper  care  is  exer- 
cised good  results  may  be  expected.  Better  results  may  be 
obtained  by  using  a  special  battery  testing  key,  or  a  pendulum 
apparatus  which  will  close  and  open  the  keys  in  precisely  the 
same  manner  each  time. 

During  the  second  hour  the  battery  will  recover  from  the 


AMMETER  AND  VOLTMETER  METHODS 


31 


effects  of  polarization,  more  or  less  completely,  and  at  the  end 
of  the  2-hour  test  the  value  of  E  should  be  about  the  same 
as  at  the  beginning.  The  recovery  curve  may  be  plotted 
backward  across  the  sheet  containing  the  other  curves,  thus 
showing  very  clearly  the  extent  of  the  recovery. 
The  data  may  be  recorded  as  below. 

TIME  TEST  OF  A  . .         . .  CELL 


Time 
hour 

of  day 
minute 

E 

E' 

I 

r 

CHAPTER  II 

BALLISTIC  GALVANOMETER  AND  CONDENSER 
METHODS 

26.  Capacity. — Elementary  Ideas. — A  given  quantity  of  air 
will  fill  a  certain  vessel  to  a  definite  pressure,  this  pressure 
depending  upon  the  amount  of  air  and  the  capacity  of  the 
vessel.  The  more  the  air  and  the  less  the  capacity  of  the 
vessel,  the  higher  the  pressure  to  which  the  former  will  be 
subjected. 

In  electricity,  when  a  conductor  is  charged  with  a  quantity 
Q,  of  electricity,  it  is  raised  to  an  electrical  pressure,  or  po- 
tential, V.  This  potential  will  be  greater  or  less  according 
to  a  property  of  the  conductor  which  is  called  its  electrical 
capacity.  The  relation  between  Q  and  V  is  expressed  by  the 
formula, 

Q  =  CV, 

where  C  denotes  the  capacity  measured  in  certain  units.  Since 
the  units  of  Q  and  V  have  been  previously  established  this  rela- 
tion shows  what  value  of  the  capacity  should  be  taken  as  unity. 
A  conductor  has  unit  capacity,  therefore,  when  unit  quantity 
of  electricity  raises  it  to  unit  potential. 

Definition. — By  the  capacity  of  a  conductor  is  meant  its 
ability  to  hold  a  quantity  of  electricity.  It  is  measured  by  the 
number  of  coulombs  per  volt  required  to  charge  the  conductor. 

When  a  body  is  charged  with  a  quantity  +  Q,  there  is  a 
complementary  quantity,  —  Q,  on  the  surrounding  surfaces. 
If  these  surroundings  are  not  far  off  the  effect  of  this  charge, 
—  Q,  will  be  to  make  the  potential  of  the  conductor  less  than  it 
otherwise  would  be.  In  other  words,  the  capacity  of  the  con- 
ductor has  been  increased  by  the  presence  of  the  other  con- 
ductors near  it.  It  will  now  require  a  larger  charge  to  raise 
it  to  the  same  potential  as  before.  As  more  water  can  be 

32 


BALLISTIC  GALVANOMETER  33 

contained  in  a  given  vessel  by  condensing  it  from  a  vapor  to  a 
liquid,  so,  by  analogy,  this  arrangement  of  conductors  whereby 
the  charge  is  increased  without  increasing  the  potential  is 
called  a  condenser.  This  name  does  not  mean  that  electricity 
is  condensed,  like  steam  into  water.  The  analogy  is  only  an 
apparent  one  and  must  not  be  pushed  too  far. 

27.  Condensers. — In  order  to  make  the  capacity  as  large 
as  possible  condensers  are  constructed  with  broad  sheets  of  tin 
foil  placed  as  near  as  possible  to  other  similar  sheets.  Actual 
contact  is  prevented  by  thin  layers  of  mica,  glass  or  paraffined 
paper.  Large  capacities  are  formed  by  building  up  alternate 
sheets  of  tin  foil  and  dielectric, 
every  other  sheet  of  tin  foil  being 
connected  to  one  terminal  post,  » 
and  the  intermediate  ones  to  the 

other  terminal  (Fig.  13). 

,„,       .  ,  .    , ,  FIG.  13. — The  tin  foil  plates 

The  best  condensers  and  those  Of  a  condenser. 

intended    for  standards  are  made 

with  thin  sheets  of  mica  as  insulation  between  the  sheets  of 
tin  foil.  Very  good  condensers  are  made  with  paper  insula- 
tion, the  whole  pressed  firmly  together  and  boiled  in  paraffine 
until  all  the  air  and  moisture  has  been  expelled,  when  the 
whole  is  allowed  to  solidify. 

A  charge  "Q"  in  a  condenser  means  that  there  is  a  charge 
of  +  Q  coulombs  on  one  set  of  plates,  and  an  equal,  though 
negative,  charge  of  —  Q  coulombs  on  the  other  set  of  plates. 
We  can  think  of  the  discharge  of  the  condenser  as  consisting  of 
the  +  Q  going  over  to  the  other  side  and  neutralizing  the  nega- 
tive charge,  thus  giving  rise  to  a  positive  current  through  the 
discharging  circuit;  or,  if  the  negative  charge  passed  through 
the  circuit  in  the  other  direction,  the  effect  would  still  be  that 
of  a  positive  current;  or,  if  we  prefer  to  think  in  terms  of  a 
"two  fluid"  theory  and  thus  consider  the  positive  charge  as 
coming  out  from  one  side  of  the  condenser  and  meeting  the 
negative  charge  as  it  comes  out  of  the  other  side,  the  two 
neutralizing  somewhere  in  the  conductor,  the  effect  is  still  the 


34  ELECTRICAL  MEASUREMENTS 

same.  This  illustrates  how  the  same  set  of  observed  facts  can 
be  explained  equally  well  in  terms  of  widely  different  theories. 
If  either  charge  is  "real,"  it  is  probably  the  -  Q,  and  the 
negative  electrons  travel  in  the  opposite  direction  to  the 
"  current." 

28.  Unit  Capacity. — In  the  international  units  the  "unit  of 
capacity  is  the  international  farad,  which  is  the  capacity  of  a 
condenser  charged  to  a  potential  of  one  international  volt  by  one 
international  coulomb  of  electricity." 

The  farad  is  thus  1,000,000,000  times  smaller  than  the  C.G.S. 
unit  of  capacity.  But  even  then  the  farad  is  far  too  large  for 
ordinary  use,  and  it  is  customary  to  express  capacities  in  terms 
of  a  smaller  unit,  the  microfarad,  which,  as  its  name  indicates, 
is  one  millionth  of  a  farad. 

Problem. — A  3-microfarad  paraffined  paper  condenser  is  about  one 
foot  square  and  an  inch  in  thickness.  How  large  a  pile  of  such  con- 
densers would  have  a  capacity  of  one  farad  ? 

How  large  a  pile  would  it  take  to  have  a  capacity  of  one  C.G.S. 
electromagnetic  unit? 

Note. — The  capacity  of  a  solid  metal  sphere  of  1  cm. 
radius  is  one  C.G.S.  electrostatic  unit,  when  it  is  far  from 
all  other  conductors. 

29.  Ballistic  Galvanometer. — A   ballistic   galvanometer   is 
one  in  which  the  moving  system,  whether  coil  or  magnet,  is 
made  comparatively  heavy  and  massive  so  that  it  will  swing 
slowly.     Such  a  galvanometer  is  designed  to  measure,  not 
steady  currents  like  an  ammeter,  but  transient  currents  which 
may  exist  for  only  a  very  small  fraction  of  a  second.     Indeed 
the  duration  is  so  short  that  it  is  customary  not  to  speak  of  them 
at  all  as  currents,  but  only  to  consider  the  total  quantity  of 
electricity  that  has  passed. 

Thus  there  are  two  reasons  for  having  a  slow  moving  gal- 
vanometer. In  the  first  place,  it  is  most  sensitive  when  in 
the  position  of  rest  and  therefore  should  not  turn  appreciably 
from  this  position  before  all  the  electricity  has  been  able  to  flow 
through  the  galvanometer  and  exert  its  full  effect  in  turning  the 


BALLISTIC  GALVANOMETER  35 

coil.  In  the  second  place,  the  coil  gives  one  kick  and  then 
settles  back  to  the  position  of  rest  again  and  the  only  thing 
that  can  be  measured  is  the  maximum  deflection  which  it 
attains.  Therefore  it  must  move  slow  enough  to  enable  one  to 
read  the  deflection  at  the  end  of  its  swing. 

For  small  deflections  the  maximum  swing  is  proportional  to 
the  quantity  of  electricity,  and  the  proportionality  factor,  or 
the  quantity  of  electricity  per  millimeter  of  deflection,  is  called 
the  constant  of  the  galvanometer.  It  is  determined  by  dis- 
charging through  the  galvanometer  a  known  quantity  and 
noting  the  resulting  deflection.  Then, 

Q-H 

where  k  is  the  desired  constant.  Knowing  k  for  a  galvano- 
meter, any  other  quantity  can  be  measured  by  sending  it 
through  the  galvanometer  and  noting  the  corresponding 
deflection. 

30.  Use  of  Ballistic  Galvanometer  and  Condenser. — When 
the  poles  of  a  battery  are  joined  to  the  plates  of  a  condenser 
the  latter  becomes  charged,  as  explained  above.  The  amount 
of  this  charge  depends  upon  the  electromotive  force,  E,  of 
the  battery  and  the  capacity,  C,  of  the  condenser  being  given 
by  the  relation 

Q  =  C  E 

When  the  condenser  is  discharged  through  a  ballistic 
galvanometer  it  will  produce  a  deflection  d,  proportional  to  Q 
or 

Q=  kd, 

where  k  is  the  constant  of  the  galvanometer. 

Ifvthis  operation  has  been  carefully  arranged  so  that  there 
has  been  no  leakage  or  discharge  of  electricity  elsewhere,  it  is 
evident  that  the  quantity  that  has  been  discharged  through  the 
galvanometer  is  the  same  quantity  that  was  put  into  the  con- 
denser by  the  battery.  That  is, 

CE  =  kd 


36 


ELECTRICAL  MEASUREMENTS 


This  is  a  very  useful  relation,  since  it  can  be  used  to  com- 
pare the  value  of  any  one  of  the  factors  involved  when  the 
other  three  are  known  or  can  be  measured. 

The  best  arrangement  for  using  a  ballistic  galvanometer 
with  a  condenser  is  shown  in  the  figure,  where  G  represents 
the  galvanometer,  C  the  condenser,  with  the  battery  at  B. 

These  are  connected  through  the 
key  as  shown.  Note  that  the  tongue 
of  the  key  is  connected  to  the  con- 
denser, and  to  the  condenser  only. 
This  precaution  is  necessary  in  or- 
der that  by  no  possibility  can  the 
battery  ever  be  joined  directly  to 
the  galvanometer.  Arranged  as 
shown,  when  the  key  is  depressed 
the  condenser  is  joined  to  the  bat- 
tery and  becomes  charged.  When 
the  key  is  raised  the  condenser  is 
joined  to  the  galvanometer  and  the 
charge  passes  through  it  producing 
FIG.  14. — Use  of  a  condenser,  a  deflection. 

31.  Damping  of  a  Galvanometer, 

Critical  Damping. — After  the  galvanometer  has  given  its  de- 
flection the  very  fact  that  the  moving  system  is  massive,  and 
at  the  same  time  can  move  freely,  which  is  essential  for  a 
good  ballistic  galvanometer,  makes  it  very  slow  in  coming  to 
rest  again.  It  will  swing  back  and  forth  many  times  until 
its  energy  has  been  used  up  in  friction  against  the  air,  and 
in  other  ways,  when  it  will  finally  settle  down  at  rest.  That 
the  swings  decrease  at  all  is  due  to  the  damping  of  the  mo- 
tion, as  this  effect  of  friction,  etc.,  is  called. 

If  the  damping  is  increased,  as  would  be  the  case  if  the  coil 
were  surrounded  with  oil,  the  swings  would  decrease  more 
rapidly  and  the  coil  would  quickly  come  to  rest.  Such 
damping  might  be  so  great  that  there  would  be  no  swings,  and 
the  coil  would  slowly  creep  back  to  the  position  of  rest, 


BALLISTIC  GALVANOMETER  37 

possibly  taking  longer  to  do  so  than  when  it  is  allowed  to 
swing  freely. 

Thus  it  is  seen  that  there  is  some  intermediate  value  of  the 
damping  that  would  allow  the  coil  to  swing  back  to  rest  not 
too  slowly  and  yet  bring  it  to  rest  without  its  swinging  to  the 
other  side.  This  value  of  the  damping  is  called  critical 
damping,  and  with  this  damping  the  coil  is  brought  to  rest  in 
the  minimum  time. 

The  most  convenient  way  to  increase  the  damping  of  a 
ballistic  galvanometer  is  to  join  its  terminals  by  a  key  or  a 
low  resistance,  as  shown  at  S,  Fig.  14.  Due  to  the  motion  of 
the  coil  in  a  strong  magnetic  field  an  induced  current  will 
flow  through  S,  when  closed,  and  the  supply  of  energy  in  the 
coil  is  quickly  dissipated  as  heat  by  the  electric  current  in  the 
wire.  Often  S  is  a  resistance  adjusted  to  such  a  value  as  to 
give  critical  damping.  The  galvanometer  can  then  be  used 
once  a  minute,  or  oftener  if  desired. 

The  effect  of  the  shunt  in  reducing  the  deflections  is  dis- 
cussed in  the  next  chapter. 

32.  The  Constant  of  a  Ballistic  Galvanometer.— For  the 
complete  theory  of  the  ballistic  galvanometer  and  a  full 
discussion  of  the  various  factors  entering  into  k,  the  reader 
is  referred  to  the  chapter  on  the  measurement  of  capacity. 
It  is  shown  in  Article  118  that 


Suffice  it  here  to  observe  that  k  is  a  constant,  resulting  from 
the  combination  of  the  various  constants  in  this  expression; 
and  without  knowing  the  value  of  any  separate  factor,  the 
value  of  k  can  be  readily  determined  as  shown  below. 

The  galvanometer,  condenser,  and  battery,  are  connected  as 
shown  in  Fig.  14.  The  scale  and  telescope  should  be  adjusted 
so  that  both  the  divisions  and  the  numbers  on  the  scale  are 
distinctly  seen  in  the  telescope.  The  eyepiece  must  be  focused 
on  the  cross  hair  of  the  telescope,  which  should  appear  very 


38 


ELECTRICAL  MEASUREMENTS 


clearly  defined.  When  focusing  the  telescope  upon  the  image 
of  the  scale  as  seen  reflected  from  the  mirror  of  the  galvan- 
ometer, it  must  be  remembered  that  the  image  is  not  on  the 
surface  of  the  mirror,  but  lies  as  far  back  of  the  mirror  as  the 
scale  is  in  front  of  it. 

After  the  setup  has  been  tried  and  found  to  work  correctly 
with  an  old  dry  cell  for  B,  a  standard  cell  whose  E.M.F.  is 
much  more  exactly  known  may  be  substituted  for  it.  Ten 
independent  determinations  of  d  should  be  made,  and  the  mean 
value  used  in  the  computation  for  k.  If  the  scale  is  movable 
it  will  be  necessary  to  measure  the  distance  of  the  scale  from 
the  mirror  of  the  galvanometer  and  compute  the  value  of  d 
for  a  standard  distance  of  1  meter. 

The  relation  given  in  Article  30  is 

CE  =  kd. 

In  the  present  case  the  condenser  used  must  be  one  of  known 
capacity.  Likewise  E  must  be  known  as  exactly  as  the  value 
of  k  is  desired.  And  d  is  measured.  Hence  the  computed 
value  of  k  is, 


If  the  value  of  C  is  given  in  microfarads,  k  will  be  expressed  in 
microcoulombs  per  scale  division.  A  microcoulomb  is  one 
millionth  of  a  coulomb  and  is  the  quantity  of  electricity  that 
is  represented  by  a  current  of  one  ampere  flowing  for  one  mil- 
lionth of  a  second.  The  data  can  be  recorded  as  follows: 


Galvanometer 

Deflection  for 

c 

E 

k 

scale  at  1  meter 

Zero 

Reading 

Deflection 

BALLISTIC  GALVANOMETER 


39 


33.  Comparison  of  E.M.F's.  by  Condenser  Method. — The 

arrangement  of  a  condenser  with  a  ballistic  galvanometer, 
may  be  conveniently  used  to  measure  the  E.M.F.  of  a  battery, 
or  any  other  difference  of  potential.  It  thus  serves  as  a  volt- 
meter, and  has  the  advantages  over  the  ordinary  voltmeter 
in  that  it  measures  the  total  E.M.F.  of  the  battery,  no  matter 
what  the  internal  resistance  of  the  latter  may  be. 

The  setup  is  arranged  as  shown  in  Fig.  15.  When  the  key 
is  depressed  the  condenser  is  charged,  and  by  raising  the  key 
it  is  discharged  through  the  galvanometer.  It  is  immaterial 
whether  the  key  works  this  way  or  whether  the  condenser  is 
charged  when  the  key  is  up  and  discharged  by  depressing  the 


FIG.  15.  —  To  measure  the  E.M.F.  of  B. 

key.     It  is  absolutely  necessary,  however,  that  the  tongue  of 
the  key  be  joined  to  the  condenser  as  shown  in  Fig.  15. 

The  relationship  given  above  is  CE  =  kd.     Solving  this  for 
E  gives,  , 


The  E.M.F.  of  the  battery  is  thus  measured  by  the  first 
throw  of  the  galvanometer  needle,  which  is  read  by  a  tele- 
scope and  scale.  The  best  way  to  determine  the  factor  k/C  is 
to  use  one  cell  of  known  E.M.F.  and  observe  the  corresponding 
deflection.  Then, 

k       E' 


E' 


fe  ,, 

=  cd> 


or 


40  ELECTRICAL  MEASUREMENTS 

so  that  finally 


Having  determined  this  coefficient  of  d  once  for  all,  the 
E.M.F.  of  any  cell  can  be  measured  quickly  and  easily  by 
observing  the  corresponding  deflection  of  the  galvanometer. 

Inasmuch  as  the  reading  must  be  caught  quickly  at  the  end 
of  the  swing  it  will  be  best  to  take  five  trials  and  use  the  mean 
deflection  for  computing  the  value  of  E.  The  data  may  be 
recorded  as  follows: 


Galvanorn  eter 

Name  of 

Mean 

ft 

E.M.F. 

cell 

Zero 

Reading 

Deflection 

deflection 

C 

of 
Cell 

34.  Comparison  of  Capacities  by  Direct  Deflection. — The 

same  arrangement  described  above  and  shown  in  Fig.  15,  can 
be  used  equally  well  for  the  measurement  of  the  capacity  of  a 
condenser.  It  is  only  necessary  to  go  through  the  experiment 
as  before,  and  observe  the  galvanometer  deflection  for  the 
relation, 

CE  =  kd. 

Now  replacing  the  condenser  by  another  one,  but  using  the 
same  battery  and  everything  else  the  same  as  before,  the 
relation  becomes, 

C'E  =  kd' 

where  df  is  the  galvanometer  deflection  when  the  condenser  of 
capacity  Cr  is  used.  Dividing  the  second  equation  by  the 
first  gives, 


BALLISTIC  GALVANOMETER 


41 


If  C  is  a  known  capacity  then  the  value  of  Cf  can  be  deter- 
mined as  exactly  as  the  flings  d  and  d!  can  be  measured. 
Each  of  these  should  be  taken  several  times,  and  the  mean 
values  used  in  the  computation. 

35.  Internal  Resistance  of  a  Battery  by  the  Condenser 
Method. — The  condenser  method  offers  a  convenient  and  ele- 
gant means  for  determining  the  internal  resistance  of  a  cell, 
the  principal  advantage  being  that  polarization  can  be  almost 
entirely  avoided.  In  the  ammeter-voltmeter  method  a  con- 


FIG.  16. — To  measure  the  resistance  of  B. 

siderable  current  must  oftentimes  be  drawn  from  the  cell  and 
for  a  period  long  enough  to  read  both  instruments.  Such 
readings  seldom  can  be  repeated,  for  owing  to  polarization  the 
cell  does  not  return  to  its  original  condition. 

The  setup  for  using  the  condenser  method  is  shown  in  the 
figure.  When  K2  is  closed  a  current  flows  through  R  and  the 
cell,  the  value  of  which  is  expressed  as 

i-   E 

B+r 


42  ELECTRICAL  MEASUREMENTS 

from  which 

E  -E' 
r  =  R^P-  (A) 

where  E'  is  written  for  RI,  the  external  fall  of  potential.     If 
R  is  known  it  only  remains  to  measure  E  and  E'. 

When  K\  is  depressed  the  condenser  is  charged  to  the  E.M.F. 
of  the  cell,  the  quantity  of  electricity  in  the  condenser  being 
Q  =  CE.  On  raising  KI  the  condenser  is  discharged  through 
the  galvanometer  which  gives  a  throw  of  the  needle  that  is 
proportional  to  the  quantity  discharged,  the  relation  being, 

Q  =  M  =  CE 
or, 

'  •„•»«, 

Similarly,  if  K\  is  worked  while  Kz  is  closed  the  potential 
difference  between  the  points  a  and  b  is  RI  or  E',  and  we  have, 


Substituting  these  expressions  in  (A)  gives, 

d-d! 


r  =  R 


d' 


It  is  best  to  use  values  of  R  that  will  give  d'  about  half  as 
large  as  d. 

In  this  way  it  is  only  necessary  to  keep  K2  closed  long 
enough  to  depress  and  raise  KI.  With  skill  this  interval  can 
be  reduced  to  less  than  a  second  when  the  keys  are  worked 
by  hand.  It  is  much  better,  however,  to  use  a  special  battery 
testing  key.  This  key  is  really  several  keys  on  one  base  and 
so  arranged  that  a  single  pressure  works  them  all  in  the  proper 
order.  In  the  present  case,  first  K%  would  be  closed,  then  K\ 
depressed  and  raised,  and  finally  Kz  opened  all  by  a  single 
downward  pressure.  This  key  is  a  regular,  three-tohgued, 
successive  contact  key  such  as  is  usually  employed  with  a 


BALLISTIC  GALVANOMETER 


43 


Wheatstone's  bridge,  modified  by  the  addition,  of  a  break 
contact  on  the  top  of  each  tongue.  (See  also  Fig.  66  for  a 
side  view  of  this  key.)  By  this  means  it  is  also  a  successive 
break  key,  the  breaks  being  alternated  with  the  makes.  Thus 
almost  any  desired  combination,  of  makes  and  breaks  can  be 
readily  obtained. 

The  arrangement  with  this  key  is  shown  in  the  figure,  where 
the  outline  of  the  key  is  drawn,  and  the  connections  indicated 
for  making  the  changes  noted  above. 
With  this  key  any  reading  can  be 
repeated  as  often  as  desired,  since 
the  cell  does  not  become  polarized 
in  the  very  short  time  the  current 
is  flowing.  By  holding  down  the 
lower  tongue  until  the  rest  of  the 
key  has  been  raised  there  will  be 
no  current  taken  from  the  cell  as 
the  key  is  being  released. 

Depressing  the  whole  key,  then, 
gives  the  deflection  d'.  Working 
the  lower  tongue,  only,  gives  d. 

Measure  in  this  way  the  internal 
resistance  of  several  cells,  making  five  or  more  determinations 
of  each  cell. 

The  data  may  be  recorded  as  follows: 


FlG"  "-- 


***• 


Name  of 
cell 

Galvanometer 

R 

r 

Zero 

Reading 

Deflection 

d 

d' 

44 


ELECTRICAL  MEASUREMENTS 


36.  Insulation  Resistance  by  Leakage.  First  Case. — This 
method  is  used  when  the  resistance  to  be  measured  is  so  large 
that  the  current  which  it  is  possible  to  pass  through  it  is  too 
small  to  be  measured  by  a  sensitive  galvanometer.  The 
method  consists,  in  brief,  in  letting  the  current  flow  into  a 
condenser  for  a  sufficient  time,  and  then  discharging  the 
accumulated  quantity  through  a  ballistic  galvanometer. 

The  setup  is  arranged  as  shown  in  Fig.  18,  where  R  denotes 
the  large  resistance  to  be  measured.  A  battery  of  sufficient 

E.M.F.,  E,  supplies  the  current 
which  flows  through  R  and  grad- 
ually charges  the  condenser  C. 
When  a  sufficient  charge  is  accu- 
mulated it  is  discharged  through 
the  galvanometer  by  closing  the 
key  K.  The  other  key  K',  is  for 
damping  the  swings  of  the  galva- 
nometer and  bringing  it  to  rest: 
and  it  should  be  kept  closed  all  the 

time  that  the  galvanometer    is  not  being  observed  through 
the  telescope. 

At  any  instant  during  the  time  the  current  is  flowing 
the  fall  of  potential  over  R  is 


FIG.  18. — To  measure  a  resist- 
ance of  many  megohms. 


RI  =  E  -  V 


(1) 


where  V  is  the  difference  of  potential  across  the  condenser. 

In  this  method  it  is  assumed  that  the  condenser  has  con- 
siderable capacity,  and  that  the  charging  is  discontinued  be- 
fore V  has  reached  an  appreciable  part  of  E.  If  the  experi- 
ment is  not  worked  in  this  way  the  following  discussion  does 
not  apply. 

At  the  start,  and  as  long  as  V  can  be  neglected  in  comparison 
with  E,  the  current  through  R  is,  from  (1), 


i-I 

1  "  R 


(2) 


BALLISTIC  GALVANOMETER  45 

If  this  current  flows  into  the  condenser  for  t  seconds,  the 
accumulated  charge  is, 

Q  =  It,  (3) 

and  when  the  condenser  is  discharged  through  the  galva- 
nometer there  is  a  deflection,  or  fling,  of  d  scale  parts,  such  that 

Q  =  kd  (4) 

Thus  the  current  is, 


and  from  (2), 


_ 
:    t 


_  E  _tE_  _  E_j^    d^ 
''  I  "  kd  ~  E'  Cf  d 


The  "constant,"  k,  of  the  galvanometer  can  be  determined 
by  the  method  described  in  Article  32. 

If  the  same  battery  is  used  in  finding  the  constant  as  in 
the  experiment  proper,  E  =  E',  and  the  absolute  value  of 
the  E.M.F.  employed  does  not  enter  into  the  computation. 

Then  R  =  ^  ~d 

37.  Insulation  Resistance  by  Leakage.  Second  Case. — If 
the  high  resistance  has  also  consid- 
erable capacity  it  will  not  be  neces- 
sary to  use  a  separate  condenser. 
Thus  if  it  is  required  to  measure 
the  insulation  of  a  condenser  or  a 
long  cable  the  arrangement  will  be 

as  shown  in  Fig.  19,  where  CR  rep- 

,11  f  ',       FIG.  19. — To  meaure  the  resis- 

resents  the  condenser  of  capacity         stance  of  a  condenser. 

C  and  resistance  R.     Upon  clos- 
ing K,  with  K'  also  closed,  the  condenser  is  charged  to  the 
fnll  potential  difference  of  the  battery.     When  the  key  is 
opened  the  charge,  Q,  leaks  through  the  high  resistance  R. 
At  the  start,  and  before  the  charge  in  the  condenser  has  been 


46  ELECTRICAL  MEASUREMENTS 

appreciably  reduced  by  the  leakage,  the  current  through  R  is, 

7  =  R' 

This  current  will  reduce  the  original  charge  in  the  condenser  by 
the  amount 

in  t  seconds,  where  t  is  not  too  great  to  consider  the  current 
constant  during  this  interval.  This  loss  of  charge  can  be  deter- 
mined by  recharging  the  condenser  through  the  galvanometer 
to  the  original  difference  of  potential.  Then  as  before, 

q  =  kd,   and   /  =  -j 

from  which 

_  E  _  Et  _    t  d' 
R  =  T   =  kd  =  C"' ¥ 

the  same  as  before  if  the  same  battey  is  used  to  determine  k. 

At  the  beginning  of  this  test  the  values  of  R  will  usually 
be  too  low  because  of  the  effect  of  "  absorption  "  by  which  a  part 
of  the  charge  disappears.  This  reduces  the  charge  in  the  con- 
denser the  same  as  though  it  had  leaked  out.  The  true  value 
of  the  resistance  will  be  obtained  only  after  several  hours,  in 
some  cases  several  days,  but  if  a  first  test  is  being  made  it  is 
well  to  determine  the  value  of  R  at  intervals  of  a  few  minutes. 
A  curve  plotted  with  the  time  of  day  for  abscissae  and  the 
corresponding  values  of  R  for  ordinates,  will  show  this  variation 
and  indicate  the  maximum  value  of  the  insulation  resistance. 

A  resistance  not  having  any  capacity  can  be  measured  by 
this  method  by  adding  a  condenser  in  parallel  with  it.  But 
in  such  a  case  the  arrangement  shown  in  Fig.  18,  would  be 
preferable. 


CHAPTER  III 
THE  CURRENT  GALVANOMETER 

38.  Description  of  a  Galvanometer. — A  galvanometer  is 
a  delicate  and  sensitive  instrument  for  the  measurement  of 
small  electric  currents.  All  galvanometers  consist  of  two 
essential  parts — a  coil  of  wire  through  which  can  flow  the 
current  to  be  measured,  and  a  permanent  steel  magnet.  In 
some  galvanometers  the  coil  is  comparatively  large  and  is 
rigidly  fixed  to  the  frame  of  the  instrument,  while  the  magnet  is 
a  small  piece  of  steel  suspended  lightly  by  a  fiber  of  untwisted 
silk  or  of  quartz.  In  other  galvanometers  the  arrangement  is 
reversed.  The  coil  is  made  as  light  as  possible  and  is  suspended 
by  a  thin  strip  of  phosphor  bronze  between  the  poles  of  a  large 
and  strong  magnet  which  often  forms  the  body  of  the  instru- 
ment. In  either  style  the  movable  portion  is  made  to  turn  as 
easily  as  possible,  the  amount  of  turning  being  measured  by  the 
mirror,  scale  and  telescope  method. 

There  are  two  ways  of  using  a  galvanometer.  A  transient 
current,  like  the  discharge  of  a  condenser,  will  produce  a  fling 
or  kick  of  the  galvanometer  after  which  it  will  settle  back  to  the 
original  position.  Evidently  the  only  thing  that  can  be 
measured  in  this  case  is  the  maximum  fling.  But  if  the 
current  is  steady  the  galvanometer  will  settle  down  at  a  de- 
flected position,  and  the  deflection,  as  the  distance  of  this 
position  on  the  scale  from  the  position  of  rest  is  called,  measures 
the  current. 

Most  galvanometers  are  so  constructed  that,  for  small  angles 
at  least,  the  deflection  is  directly  proportional  to  the  current. 
That  is,  I  =  Fd. 

The  proportionality  factor,  F,  is  called  the  " figure  of  merit'\ 
of  the  galvanometer,  and  it  is  defined  as  the  current  per  scale 

47 


48 


ELECTRICAL  MEASUREMENTS 


division  (1  mm.)  that  will  deflect  the  galvanometer.  The 
figure  of  merit  of  most  galvanometers  is  smaller  than  one 
hundred-millionth  of  an  ampere  per  millimeter. 

Inasmuch  as  the  deflection  will  vary  with  the  distance  of  the 
scale  from  the  mirror,  this  distance  should  be  made  1  meter. 
If  for  any  reason  the  scale  is  at  a  different  distance  the  observed 
deflection  must  be  corrected  to  what  it  would  have  been  had 
the  scale  been  at  the  proper  distance. 

39.  Figure  of  Merit,  (a.)  By  Direct  Deflection. — In  order  to 
determine  the  figure  of  merit  it  is  necessary  to  send  a  small 
known  current  through  the  galvanometer 
and  observe  the  steady  deflection  it  pro- 
duces. The  method  can  be  understood  by 
reference  to  Fig.  20.  The  galvanometer  is 
joined  in  series  with  a  battery,  a  large  re- 
sistance and  a  key.  When  the  key  is 
closed,  the  current  flowing  through  the  cir- 
cuit, and  therefore  through  the  galvanom- 
eter, is,  by  Ohm's  law, 

E 


R 


FIG.  20— To  deter-  ^  = 

mine  the  figure   of 
merit  of  G.  t 

where  E  denotes  the  E.M.F.  of  the  battery, 

and  g,  b,  R,  the  resistances  of  the  galvanometer,  battery,  and 
R,  respectively.  If  this  current  produces  a  steady  deflection 
of  d  scale  divisions  (mm.),  the  figure  of  merit  is  F  =  i/d. 
By  using  a  voltmeter  to  measure  directly  the  fall 
potential  between  a  and  6,  the  current  i  will  be  given  by 

.  _       V 
~  R  +  g 

where  V  is  the  voltmeter  reading.     Then 


of 


F  = 


V 


Thus  by  the  use  of  the  voltmeter,  the  somewhat  uncertain 
E.M.F.  of  the  cell  is  replaced  by  the  definite  voltmeter  reading, 


THE  CURRENT  GALVANOMETER 


49 


and  the  unknown  resistance  of  the  cell  does  not  appear  in  the 
equations.  Of  course  V  and  d  must  be  simultaneous  values, 
and  it  is  understood  that  d  is  the  steady  deflection  produced  by 
the  steady  current  i. 

40.  Figure  of  Merit.  (6)  By  Fall  of  Potential.— With  a 
sensitive  galvanometer  it  is  usually  not  possible  to  make  R 
large  enough  to  use  the  simple  method.  It  is  then  most  con- 
venient to  use  a  value  for  E  which  is  only  a  small  fraction  of  the 
E.M.F.  of  the  cell.  This  can  be 
done  by  the  fall  of  potential 
method  shown  in  Fig.  21.  The 
fall  of  potential  between  a  and 
b  is  now  PI  instead  of  E,  and 
both  P  and  I  can  be  made  as 
small  as  necessary.  For  most 
galvanometers  it  is  convenient 
to  make  P  +  Q  =  1000  ohms, 
and  R  about  100,000  ohms. 

The  current  from  the  battery 
divides  at  a,  one  part,  i,  going 

through   the  galvanometer,  an-    ^     01      ^. 

T  f  FIG.  21.— Figure  of  merit  by  fall 

other  part,  /,  through  P,  and  the  Of  potential  method. 

third    part    flows   through    the 

voltmeter.     The  fall  of  potential  from  a  to  6  across  P  is  the 

same  as  through  R  and  the  galvanometer,  or 


PI  =  R'i 


(D 


where  Rr  is  the  combined  resistance  of  R  and  the  galvanometer 
with  its  shunt,  if  one  is  used. 

The  current  through  Q  is  I  +  i,  and  the  fall  of  potential 
from  a  to  c,  which  is  measured  by  the  voltmeter  is 

PI  +  Q(I  +  i)  =  V  4  (2) 

Eliminating  I  between  (1)  and  (2),  and  solving  for  i, 

. PV 

~  QR'  +  PR'  +  PQ 


50 


ELECTRICAL  MEASUREMENTS 


Then  as  before, 


PV 


(P  +  Q)  R'  +  PQ 


Usually  the  term  PQ  can  be  omitted  in  comparison  with 
QR',  since  P  is  much -less  than  Rf,  and  by  keeping  P  small 
the  error  thus  introduced  may  be  made  less  than  the  error  in 
reading  the  voltmeter  or  the  galvanometer  deflection.  The 
use  of  the  shunt  S,  is  to  protect  the  galvanometer  from  large 
and  unexpected  deflections.  It  should  be  kept  on  the  "zero 
position"  while  setting  up  or  taking  down  the  connections. 
It  is  always  a  wise  precaution  not  to  connect  in  the  battery 
until  after  the  rest  of  the  setup  is  completed  and  has  been  care- 
fully examined  to  make  sure  that  no  unintended  connections 
have  been  made. 

Record  data  as  follows: 

To  DETERMINE  THE  FIGURE   OF  MERIT  OP    GALVANOMETER   No. 


Zero 

Reading 

Deflection 

R 

V 

P 

Q 

F 

41.  Other  Constants  of  the  Galvanometer.  Megohm  Sen- 
sibility.— The  sensitiveness  of  a  galvanometer  is  often  expressed 
in  megohms  (millions  of  ohms).  This  means  the  number  of 
megohms  that  can  be  placed  in  series  with  the  galvanometer 
to  give  a  deflection  of  one  scale  division  per  volt  of  the  applied 
E.M.F.  Expressed  in  ohms,  its  numerical  value  is  evidently 
given  by  the  reciprocal  of  the  figure  of  merit,  so  when  the 
latter  is  given  the  megohm  sensibility  is  readily  computed. 

Voltage  Sensibility. — For  some  purposes  it  is  necessary  to 
know  the  sensitiveness  of  a  galvanometer  to  differences  of 
potential  applied  to  its  terminals.  This  will  depend  largely 
upon  the  resistance  of  the  galvanometer,  as  a  small  resist- 
ance will  allow  a  larger  current  to  flow.  The  voltage  sen- 


THE  CURRENT  GALVANOMETER 


51 


sibility  is  expressed  in  volts  per  scale  division,  and  is  given 
by  the  product  of  the  figure  of  merit  and  the  resistance  of  the 
galvanometer. 

42.  Use  of  Shunts.  Common  Form. — When  the  current  to 
be  measured  by  any  instrument  is  larger  than  the  range  of 
the  scale  the  latter  can  be  increased  to  almost  any  desired 
extent  by  placing  a  shunt  in  parallel  with  the  instrument,  as 
shown  in  Fig.  22.  The  shunt  and  instrument  thus  form  two 
branches  of  a  divided  circuit  and  the  current  through  one 
branch  is  directly  measured.  Knowing 
the  current  in  this  branch  the  total  cur- 
rent can  be  computed. 

Thus  let  G  denote  the  galvanometer 
or  ammeter,  and  S  the  shunt.     The  cur- 
rent   through  the  galvanometer  is,  by     M  — \AAA/ — 
Ohm's  law, 


N 


FIG,  22.— Use  of  a 
shunt. 


where  V  is  the  potential  difference  be- 
tween M  and  N,  and  g  denotes  the  resistance  of  the  galvanom- 
eter. In  the  same  way  the  total  current,  which  flows  through 
g  and  s  in  parallel,  is 


r  _,• 

~  gs  -*~r" 

g  +  s 

The  ratio,  I/i,  of  the  main  current  to  that  part  which  is 
measured  by  the  galvanometer  is  called  the  "multiplying 
power  of  the  shunt."  From  the  relation  above  it  is  seen  to 

be  equal  to  --  *     It  is  the  factor  by  which  the    current 
s 

measured  by  the  galvanometer  must  be  multiplied  to  give  the 
total  current  through  the  main  circuit.  In  order  that  this 
factor  may  be  expressed  in  convenient  round  numbers,  10, 
100,  1000,  etc.,  it  is  necessary  to  have  a  series  of  shunts  care- 


52  ELECTRICAL  MEASUREMENTS 

fully  adjusted  to  g>  gg>  ggg,  etc.,  of  the  resistance  of  the  gal- 
vanometer. Such  shunts  will  not  have  the  same  multiply- 
ing power  when  used  with  a  galvanometer  of  different  resist- 
ance, and  therefore  can  be  used  advantageously  only  with 
the  galvanometer  for  which  they  were  made.  Placing  a  shunt 
around  a  galvanometer  will  reduce  the  total  resistance  of  the 
circuit  and  therefore  the  current  measured  by  the  galvanom- 
eter times  the  multiplying  power  of  the  shunt  does  not  give 
the  value  of  the  original  current,  but  the  value  of  the  new 
main  current.  Sometimes  extra  resistances  of  0.90,  0.990, 
and  0.9990,  are  inserted  to  keep  the  total  resistance  of  the 
circuit  constant. 

When  the  galvanometer  is  used  ballistically  these  shunt 
ratios  are  not  the  same  as  for  steady  currents  because  of  the 
varying  amounts  of  damping  produced. 

43.  Universal  Shunt. — Another  way  of  using  a  shunt  is  to 
connect  a  resistance  as  a  permanent  shunt  across  the  ter- 
minals of  the  galvanometer.  The  current  to  be  measured  is 
passed  through  only  a  part  of  this  shunt  the  remainder  acting 
merely  as  resistance  in  series  with  the  galvanometer.  Fig. 
23  shows  a  four-step  shunt,  the  total  resistance  of  all  being  S, 
which  is  joined  as  a  permanent  shunt  on  the  galvanometer. 
The  multiplying  power  of  this  arrangement  is 

(g  +  S  -  s)  +  s        g  +  S 


where  S  is  always  the  same  constant  resistance.  Therefore 
in  changing  from  one  shunt  to  another  the  numerator  remains 
constant  and  the  change  in  multiplying  power  depends  only 
upon  the  changes  made  in  s.  Furthermore,  no  error  is  intro- 
duced by  contact  resistance  at  b  as  any  resistance  at  this 
place  does  not  affect  the  accuracy  of  the  shunt  ratios.  The 
total  resistance  of  the  shunt  should  be  about  twenty  times  that 
of  the  galvanometer. 

A  universal  shunt  box  consists  of  several  coils  permanently 


THE  CURRENT  GALVANOMETER 


53 


joined  in  series.  When  used  as  a  shunt  the  galvanometer 
terminals  are  connected  to  the  extremities  of  this  series,  thus 
using  the  entire  resistance.  The  current  is  not  passed  through 
all  of  the  resistance  in  the  shunt  box  as  in  the  case  of  a  common 
shunt,  but  it  may  be  passed  through  either  one  or  several  of 
the  coils  constituting  the  series.  Thus  the  part  which  actu- 
ally carries  the  current  is  the  real  shunt,  while  the  remaining 
coils  are  thrown  in  with  the  galvanometer  to  further  aid  in 


-5=1000  Ohms- 


90  goo 


C 


0                       .001  .01  .1 
S=  Shunt  Resistance— 


FIG.  23. — Diagram  illustrating  the  universal  shunt. 

reducing  the  deflection.  The  arrangement  will  be  clearly 
understood  by  referring  to  the  figure,  which  shows  the  connec- 
tions of  a  universal  shunt  having  a  total  resistance  of  1000 
ohms.  The  figure  shows  the  shunt  set  for  a  multiplying  power 
of  100.  GI,  <T2,  BI,  Bz,  are  binding  posts  where  connections 
to  the  external  circuits  are  made. 

The  change  produced  in  the  resistance  of  the  total  circuit 
is  not  as  easily  determined  as  for  a  common  shunt.     Indeed 


54  ELECTRICAL  MEASUREMENTS 

the  resistance  is  frequently  greater  with  the  shunt  than  when 
the  galvanometer  is  used  alone.  In  many  kinds  of  work  it  is 
not  essential  that  the  resistance  shall  be  constant  or  even 
known.  Where  it  must  be  known  it  can  be  determined  for  the 
galvanometer  and  its  shunt  combined  as  readily  as  for  the 
galvanometer  alone. 

Shunts  are  marked  with  the  numbers  0.1,  0.01,  0.001,  imply- 
ing the  fractions  of  the  current  which  they  pass  through  the 

galvanometer,  or  with  the  numbers  <r  >  ^  -^^  implying  the 

ratio  between  their  resistance  and  that  of  the  galvanometer. 

It  is  evident  that  when  the  universal  shunt  is  used  at  the 
point  marked  1.  the  galvanometer  is  not  quite  as  sensitive  as 
with  no  shunt  connected.  If  S  is  several  times  g  this  slight 
reduction  in  the  sensitiveness  is  of  small  moment.  The 
essential  thing  is  that  when  the  shunt  is  set  at  0.1,  0.01,  etc., 
the  same  total  current  will  give  deflections  0.1,  0.01,  etc.,  as 
large  as  with  the  shunt  set  at  1.  And  this  arrangement  of 
the  galvanometer  shunts  is  especially  useful  because  a  given 
series  of  shunts  will  have  the  same  relative  multiplying 
powers  when  used  with  any  galvanometer.  Since  the  damp- 
ing is  constant,  the  shunt  ratios  remain  the  same  when  the 
galvanometer  is  used  ballistically. 

Any  ordinary  resistance  box  having  a  traveling  plug  for 
making  a  third  connection  at  any  intermediate  point  can  be 
used  as  a  universal  shunt  for  any  galvanometer.  I?or  all 
values  the  shunt  ratios  are  very  accurate  since  all  the  coils 
are  even  ohms  and  can  be  adjusted  much  more  precisely 
than  in  the  case  of  common  shunts.  Differences  in  tempera- 
ture between  the  galvanometer  and  shunt  produce  no  error, 
but  should  remain  constant  while  measurements  are  being 
made. 

44.  The  Multiplying  Power  of  a  Shunt.  Test  of  a  Shunt 
Box. — The  effect  of  a  shunt  in  increasing  the  amount  of 
current  that  can  be  measured  by  a  galvanometer  can  be 
determined  experimentally  by  finding  the  figure  of  merit  of 


THE  CURRENT  GALVANOMETER  55 

the  galvanometer  alone,  as  shown  in  Article  40  and  then  re- 
determining  it  for  the  galvanometer  with  its  shunt,  considering 
both  together  as  a  new  instrument  having  a  resistance  of 

C\  Q 

~- —  The  ratio  of  these  two  figures  of  merit  is  the  multiply- 
9  i  s 

ing  power  of  the  shunt. 

In  the  same  manner  all  of  the  shunts  in  the  shunt  box  should 
be  tested,  and  the  multiplying  power  of  each  one  determined. 
The  results  should  be  compared  with  the  values  stamped  on 
the  shunt  box,  and  also  with  the  computed  values  as  deter- 
mined by  the  relative  resistances  of  the  galvanometer  and  the 
shunt. 

When  using  the  shunts  of  lowest  resistance  it  may  be  better 
to  determine  the  figure  of  merit  by  the  direct  deflection  method, 
Article  39.  The  scale  deflection  should  be  about  the  same 
for  each  shunt. 

45.  Resistance  of  Galvanometer  by  Half  Deflection,  (a) 
Resistance  in  Series. — In  some  of  the  foregoing  exercises  it 
is  necessary  to  know  the  resistance  of  the  galvanometer  as  it 
has  been  used,  either  alone  or  combined  with  a  shunt.  If 
this  is  unknown  it  can  be  determined  with  a  fair  degree  of 
accuracy  by  the  method  of  half  deflection. 

Let  the  galvanometer  be  connected  to  a  source  of  small 
potential  difference,  of  such  amount  that  a  large  deflection 
can  be  obtained  with  only  the  galvanometer  resistance,  G, 
in  the  galvanometer  branch.  Now  add  enough  resistance, 
R,  in  series  with  the  galvanometer  to  make  the  deflection  ex- 
actly one-half  of  its  former  value.  This  means  that  the  current 
has  been  reduced  to  half  its  former  value  and  therefore  the 
resistance,  R  +  G,  in  the  galvanometer  circuit  has  now  been 
made  twice  as  much  as  when  the  galvanometer  was  used  alone. 
That  is, 

R  +  G  =  2G          or        G  =  R. 

To  avoid  the  errors  arising  from  thermal  currents,  etc.,  it 
is  best  to  reverse  the  battery  and  repeat  the  measurements, 
taking  the  mean  of  the  two  results  as  the  correct  value  of  G. 


56  ELECTRICAL  MEASUREMENTS 

46.  Resistance  of  a  Galvanometer  by  Half  Deflection,     (b) 

Resistance  in  parallel. — When  the  resistance  of  the  galvan- 
ometer is  low,  or  a  small  E.M.F.  is  not  readily  available,  the 
galvanometer  may  be  placed  in  series  with  a  battery  of  one  or 
more  cells  and  sufficient  resistance  to  give  a  fairly  large  deflec- 
tion. Then  let  a  resistance  box  be  joined  in  parallel  with  the 
galvanometer,  and  the  resistance  varied  until  the  deflection  is 
just  half  of  its  former  value.  The  current  is  now  divided  be- 
tween the  galvanometer  and  its  shunt,  half  of  the  original 
current  flowing  through  the  galvanometer  and  the  rest  through 
the  shunt.  If  the  main  current  is  unchanged,  this  means  that 
the  current  is  equally  divided  between  the  galvanometer  and 
its  shunt.  From  this  it  follows  that  the  resistance  of  the  gal- 
vanometer is  equal  to  the  resistance  of  the  shunt. 

It  is  true  that  the  addition  of  the  shunt  has  reduced  the 
resistance  of  this  portion  of  the  circuit,  but  as  this  is  only 
a  small  part  of  the  total  resistance  in  the  battery  circuit, 
the  main  current  in  the  second  case,  and  which  is  divided 
between  the  galvanometer  and  the  shunt,  will  be  larger  than 
the  current  in  the  first  case  by  less  than  can  be  read  on  the 
galvanometer  scale.  This  point  can  be  tested  by  placing  the 
shunt  resistance  in  series  with  the  galvanometer  and  noting 
whether  it  produces  an  appreciable  effect  in  reducing  the 
deflection. 

Problem. — Let  the  student  draw  a  setup  illustrating  this  method. 
Let  him  give  a  mathematical  proof  that  G  —  S. 

47.  Differential     Galvanometer. — A     differential     galvan- 
ometer has  two  independent  coils,  as  nearly  alike  as  possible. 
A  current  passed  through  either  one  will  produce  a  deflection, 
and  if  the  current  flows  through  both  coils  in  series  the  deflec- 
tion is  due  to  the  effect  of  both  coils  acting  together.     If  the 
effect  of  the  current  in  one  coil  acting  alone  is  to  produce  a 
deflection  in  one  direction,  and  the  effect  of  a  current  in  the 
other  coil  is  to  produce  a  deflection  in  the  other  direction,  the 
effect  of  both  coils  acting  together  will  be  the  difference  of  the 


THE  CURRENT  GALVANOMETER 


57 


two,  and  the  resulting  deflection  will  be  smaller  than  for 
either  coil  alone.  If  the  coils  have  been  carefully  made, 
and  adjusted  so  that  the  magnetic  effect  of  each  upon  the 
needle  is  the  same,  then  no  deflection  will  be  produced  by  equal 
currents  flowing  through  the  two  coils  in  opposite  directions; 
for  the  effect  of  one  coil  is  just  neutralized  by  the  other. 
Usually  this  balance  is  not  exact,  and  a  final  adjustment 
is  required  before  using  the  galvanometer.  And  since  it  is 
impossible  to  have  the  coils  exctly  alike,  the  two  currents 
will  not  be  equal  for  a  balance. 

Calling  the  current  through  one  coil  I',  and  that  through 
the  other  I",  the  relation  between  them  would  be, 

If  =  nl",  (I) 

where  n  is  a  constant  whose  value  is  about  unity. 

To  compare  two  resistances  R  and  X,  they  are  joined  in 

parallel  with  each  other,  as  shown 
in  the  figure.  In  series  with  each 
is  one  coil  of  the  galvanometer. 
A  resistance,  P,  is  placed  in  series 
with  the  battery  in  order  to  keep 
the  current  from  being  excessive. 

The  galvanometer  is  adjusted  as 
follows.  R  and  X  are  short  cir- 
cuited by  inserting  all  of  the  plugs, 
or  otherwise.  This  is  better  than 
removing  them  from  the  circuit, 
as  no  change  in  connections  will 
be  required  when  they  are  to  be  used.  On  now  closing  the 
key  the  current  through  the  galvanometer  coils  will  depend 
only  upon  their  relative  resistances,  and  probably  these  are 
not  such  as  will  give  a  balance.  Let  a  resistance,  r,  be  now 
added  to  the  side  having  too  large  a  current  and  adjusted  to 
give  a  balance.  Then,  from  (1), 

V  V 

T  =  »iTI  (2) 


FIG.  24.— Use  of  the  differen- 
tial galvanometer,  A  B. 


58  ELECTRICAL  MEASUREMENTS 

where  A  and  B  are  the  resistances  of  the  two  coils,  and  V  is 
the  fall  of  potential  between  a  and  6. 

Inserting  R  and  X  ,and  adjusting  R  till  a  balance  is  again 
obtained,  gives,  from  (1), 

V  V' 

A  +  R  =  n(B+r)+X 

where  V'  is  the  new  value  of  V. 

Exchanging  R  with  X,  and  readjusting  R  to  again  balance 
gives 

7"  V'  ,* 

A  +  X  =   n(B  +  r}+R> 

where  R'  and  V"  are  the  slightly  changed  values  of  R  and  V. 
Dividing  (2)  by  (3)  gives, 

A  +  R     (B  +r)  +  X 
A  (B+r) 

Clearing  of  fractions  and  solving  for  X 

X  =  —  L-7 —  (5) 

Similarly  from  (2)  and  (4) 

*  =  w£         -  <•> 

Multiplying  (5)  and  (6)  and  extracting  the  square  root 

X  =   Vfl/P 

47A.  Differential  Galvanometer  in  Shunt. — The  differen- 
tial galvanometer  is  even  more  useful  in  the  comparison  of 
two  low  resistances.  In  this  case  each  of  the  galvanometer 
coils  is  used  as  a  sensitive  voltmeter  to  measure  the  fall  of 
potential  over  the  two  resistances.  If  the  fall  of  potential 
over  each  of  the  two  resistances  is  the  same,  when  they  are 
in  series,  the  resistances  must  be  equal. 

The  galvanometer  coils  should  be  of  high  resistance,  say 


THE  CURRENT  GALVANOMETER 


59 


several  thousand  ohms.  This  means  that  they  will  have  a 
great  many  turns  of  (copper)  wire,  which  will  make  the  galvan- 
ometer sensitive  to  small  currents.  Therefore  only  a  little 
current  will  be  shunted  from  the  low  resistances  that  are 
being  compared.  . 

For  this  comparison  it  is  necessary  to  have  a  standard  low 
resistance  which  can  be  varied  by  small  steps.  This  is  joined 
in  series  with  the  unknown  resistance  X,  as  shown  in  Fig.  25. 
Some  additional  resistance,  P,  is  placed  in  the  battery  circuit 
to  keep  the  current  from 
being  too  large.  The  two 
coils  of  the  galvanometer  are 
connected  as  shown. 

To  adjust  the  galvanom- 
eter the  two  coils  are  con- 
nected together  in  parallel 
and  both  shunted  across  the 
same  low  resistance.  Both 
coils  will  now  have  the  same 
fall  of  potential  and  the  gal- 
vanometer should  give  no 
deflection.  In  case  there  is  a  deflection,  the  current  through 
one  of  the  coils  must  be  reduced  by  adding  some  resistance, 
r,  in  series  with  this  coil  until  the  deflection  is  brought  to 
zero.  This  will  make  the  resistances  of  the  two  shunts 
unequal,  but  the  deflection  will  be  zero  when  each  shunt  has 
the  same  fall  of  potential. 

When  connected  as  shown  in  Fig.  25,  a  balance  will  be  ob- 
tained when  R  has  been  adjusted  to  equal  the  value  of  X, 
provided  that  the  current  through  each  is  the  same. 
In  general  this  will  not  be  the  case,  for  by  introducing  r  the 
shunt  currents  are  not  the  same.  Therefore  a  balance  is 
obtained  when  the  resistance  of  X  with  its  shunt  equals  the 
resistance  of  R  with  its  shunt.  That  is,  when, 

XB  RA 

X  +  B  "=  R  +  A 


FIG.  25. — Differential  galvanometer, 
A  B,  in  shunt. 


60  ELECTRICAL  MEASUREMENTS 

A  second  balance  is  obtained  by  exchanging  X  and  R,  and 
readjusting  the  value  of  the  latter  to  R'  for  zero  deflection. 
Then, 

R'B  XA 

R'  +  B  ~  X  +  A 

Dividing  the  first  of  these  equations  by  the  second  gives, 
X(Rr  +  B)         R(X  +  A) 
R'(X  +  B)     ~X(R  +  A) 
or, 

X  X       (X  +  A)  (X  +  B) 
R'R--  (R  +  A)  (#  +  *)  ==  l>  very  nearly' 
Even  in  the  extreme  case  of  A  =  1000  and  B  =  2000,  with 
X  =  1  ohm,  this  ratio  differs  from  unity  by  only  about  two  ten- 
thousandths,  and  the  difference  is  correspondingly  less  when 
A  and  B  are  more  nearly  equal. 
Therefore, 

X  = 


The  value  of  X  is  thus  a  mean  proportional  between  the 
values  of  R  required  to  give  the  two  balances.  If  these  values 
are  nearly  equal,  no  appreciable  error  is  made  by  taking  X  as 
the  arithmetical  mean.  In  case  a  change  of  one  ohm  in  R 
produces  a  readable  deflection  the  tenths  of  ohms  can  be  ob- 
tained by  interpolation. 


CHAPTER  IV 
THE  WHEATSTONE  BRIDGE 

48.  The  Wheatstone  Bridge. — The  Wheatstone  bridge 
consists,  essentially,  of  two  circuits  in  parallel  and  through 
which  an  electric  current  can  flow.  Let  these  circuits  be 
represented  by  ABD  and  ACD,  Fig.  26,  and  let  the  currents 
through  the  two  branches  be  denoted  by  /  and  /'.  Since  the 
fall  of  potential  from  A  to  D  is  the  same  whichever  path  is 


FIG.  26. — Principle  of  the  Wheatstone  Bridge. 

considered,  there  must  be  a  point  C  on  one  circuit  which  has 
the  same  potential  as  any  chosen  point  B  on  the  other.  If 
one  terminal  of  a  galvanometer  is  joined  to  B  and  the  other 
terminal  is  moved  along  ACD  the  galvanometer  will  indicate 
zero  deflection  when  the  point  C  has  been  found.  Since  B 
and  C  have  the  same  potential,  the  fall  of  potential  from  A  to 

61 


62  ELECTRICAL  MEASUREMENTS 

B  is  the  same  as  from  A  to  C,  or  in  terms  of  the  currents  and 
resistances, 

IP  =  I'Q 

where  P  and  Q  are  the  resistances  of  AB  and  AC,  respectively. 
Similarly  for  the  other  part  of  the  circuits 

IR  =  rx. 

Dividing  one  equation  by  the  other  eliminates  the  unknown 
currents  and  gives 

^=£ 
R      X 

as  the  relationship  of  the  resistances  when  the  bridge  is  bal- 
anced. In  the  usual  method  of  using  the  Wheatstone  bridge 
three  of  these  resistances  are  known  and  the  value  of  the  fourth 
is  easily  computed  from  the  above  relation  as  soon  as  a  balance 
is  obtained. 

49.  The  Slide  Wire  Bridge— Simple  Method.— The  Wheat- 
stone  bridge  principle  is  used  in  several  forms  of  apparatus 
for  the  measurement  of  resistance.  The  simplest  of  these  is 
the  slide  wire  bridge  as  shown  in  Fig.  27.  The  unknown 
resistance  which  is  to  be  measured  is  placed  at  X,  while  at  R  is 
the  known  resistance,  usually  a  box  of  coils.  The  branch 
ACD  consists  of  a  single  uniform  wire,  usually  one  meter  in 
length,  stretched  alongside  or  over  a  graduated  scale.  The 
balance  is  obtained  by  moving  the  contact  C  along  the  wire 
until  a  point  is  found  for  which  the  deflection  of  the  galva- 
nometer is  zero  when  Kf  and  K  are  closed.  This  contact  should 
not  be  scraped  along  the  wire,  but  always  raised,  moved  to  the 
new  point  and'  then  gently  but  firmly  pressed  into  contact 
with  the  wire.  Neither  should  it  be  used  for  a  key  as  the 
continual. tapping  will  dent  the  wire  and  destroy  its  uniformity. 
The  two  keys  K'  and  K  are  both  combined  into  a  single  succes- 
sive contact  key  often  called  a  Wheatstone  bridge  key,  in 
which  one  motion  of  the  hand  will  first  close  the  battery  key, 
and  then,  after  the  currents  have  been  established,  will  close 


THE  WHEATSTONE  BRIDGE 


63 


the  galvanometer  circuit.  The  need  of  such  a  key  is  very 
evident  when  there  is  self  inductance  in  X. 

When  the  point  C  has  been  located  we  have,  from  Kirch- 
hoff's  second  law,1 

xF  =  apl 

where  a  is  the  length  of  the  bridge  wire  from  A  to  C,  and  p  is  the 
resistance  of  unit  length  of  this  wire,  x  denotes  the  value  of 
the  resistance  of  X.  Similarly, 


and  dividing 


RF  =  bpl 


p    _     _  _ 
Kb~  ^1000  - 


if  the  total  length  of  the  bridge  wire  is  1000  millimeters. 
Measure  in  this  way  the  resistances  of  two  or  more  coils. 


FIG.  27. — The  simple  slide  wire  bridge. 

Also,  measure  the  same  coils  when  joined  in  series  and  compare 
the  result  with  the  computed  value, 

R  =  R'  +  R". 

When  two  coils  are  joined  in  parallel  the  measured  resistance 
should  fulfill  the  relation, 

l/R  =  1/R'  +  l/R". 
page  101. 


64 


ELECTRICAL  MEASUREMENTS 


Problem  1. — Exchange  the  positions  of  the  battery  and  the  galva- 
nometer and  then  deduce  the  formula  for  x,  as  above. 
Problem  2. —  Prove  that  for  three  resistances  in  series 

R  =  Ri  +  Rz  +  #3 
and  in  parallel 


R  = 


Problem  3. — Deduce  the  corresponding  expressions  for  five  resist- 
ances. 

50.  Calibration  of  the  Slide  Wire  Bridge. — In  deducing  the 
formula  for  the  slide  wire  bridge  it  was  assumed  that  the  bridge 
wire  was  divided  into  1000  parts  of  equal  resistance,  and  that 
the  readings  obtained  from  the  scale  corresponded  to  these 
divisions.  To  make  sure  that  the  scale  readings  do  thus 
correspond  to  the  bridge  wire  it  is  necessary  to  calibrate  the 

wire,  that  is,  to  determine 
experimentally  what  read- 
ings on  the  scale  correspond 
to  the  1000  equiresistance 
points  on  the  wire. 

Two    well-adjusted    resist- 
ance   boxes    are    inserted  in 


FIG.  28. — Calibration  of  the  sim- 
ple bridge. 


the  two  back  openings  of 
the  bridge,  and  the  battery 
and  galvanometer  connected 
in  the  usual  manner.  If  now,  for  example,  500  ohms  are  put 
in  each  box  the  balance  point  will  be  at  the  middle  of 
the  bridge  wire  and  this  should  be  at  the  point  marked  500 
on  the  scale.  If  it  is  not,  but  falls  a  distance  /  below  500, 
then  /  is  the  correction  which  must  be  added  to  the  observed 
reading  to  obtain  the  true  reading.  Since  there  may  be 
thermal  currents  in  the  galvanometer  circuit  this  value  of  / 
should  be  computed  from  the  mean  of  two  readings,  one 
taken  with  the  battery  current  direct  and  the  other  taken  with 
the  battery  reversed. 

In  the  same  way  the  true  location  of  the  points  100,  200, 
300,  400,  500,  600,  700,  800,  and  900  can  be  found.     It  will  be 


THE  WHEATSTONE  BRIDGE 


65 


found  convenient  to  keep  the  sum  of  P  and  Q  always  constant 
at  1000  ohms. 

Finally  a  calibration  curve  is  drawn,  with  the  readings  of 
the  scale  for  abscissae  and  the  corresponding  corrections  for 
ordinates.  The  corrections  for  any  point  on  the  scale  can  then 
be  read  directly  from  the  curve.  The  corrected  readings  are 
thus  expressed  in  thousandths  of  the  total  bridge  wire,  in- 
cluding all  the  resistance  of  straps,  connections,  etc.,  between 
the  two  points  where  the  battery  is  attached. 

CALIBRATION  OF  BRIDGE  No.  . 


P 

Q 

Scale  readings 

True 
reading 

Cor. 

Battery 
direct 

Battery 
reversed 

Mean 

61.  Double  Method  of  Using  the  Slide  Wire  Bridge.— In 

the  simple  method  above  only  a  single  balance  point  was 
obtained,  and  the  value  of  the  unknown  resistance  was  com- 
puted from  the  relation, 

x  _          a 

R  =  1000  -  a 

where  a  denotes  the  reading  on  the  scale  at  the  point  of 
balance,  and  is  assumed  to  be  the  length  of  one  portion  of  the 
bridge  wire. 

The  measurement  of  resistance  will  be  more  precise  if  x  and 
R  are  exchanged  with  each  other,  without,  however,  changing 
the  value  of  either  one,  and  a  new  balance  point  determined. 
This  second  balance  point  will  be,  say,  at  a'  on  the  scale,  and 

x        1000  -a' 
R  =          a' 

5 


66  ELECTRICAL  MEASUREMENTS 

Combining  these  two  equations  by  the  addition  of  pro- 
portions, 

x       1000  +  (a  -  a'}        1000  +  d 
R  ==  1000  -  (a  -  a')  ==  1000  -  d 

in  which  the  actual  values  of  a  and  a'  do  not  appear,  but  only 
their  difference.  Thus  all  questions  regarding  the  starting 
point  of  the  scale  or  the  wire  are  eliminated,  and  if  d  is  small 
any  error  made  in  its  determination  will  have  only  a  small 
effect  upon  the  value  of  x  as  computed  from  this  equation. 

52.  The  Wheatstone  Bridge  Box.  —  In  the  slide  wire  form 
of  the  Wheatstone  bridge  the  balance  is  obtained,  by  locating 
a  certain  point  on  the  wire,  and  the  accuracy  of  the  measure- 
ment depends  upon  the  accuracy  with  which  the  lengths  of 
the  two  portions  of  the  wire  can  be  measured.  In  the  Wheat- 
stone  bridge  box  the  wire  is  replaced  by  a  few  accurately 
adjusted  resistance  coils.  Thus  while  the  number  of  ratios 
that  can  be  employed  is  less  than  ten,  the  values  of  these  few 
ratios  are  precise  even  when  the  ratio  is  far  from  unity.  The 
usual  arrangement  is  to  make  P  and  Q,  Fig.  26,  the  two  ratio 
arms  with  the  unknown  resistance  in  X  and  obtain  the  balance 
of  the  bridge  by  adjusting  the  resistance  of  R.  The  value 
of  the  unknown  is  then  given  by  the  usual  relation, 


and  is  known  as  accurately  as  are  the  values  of  P,  Q  and  R. 

In  a  common  form  of  the  Wheatstone  bridge  box,  P  and  Q 
each  contain  1,  10,  100,  and  1000  ohm  coils  thus  giving  ratios 
of  1000,  100,  10,  1,  0.1,  0.01,  0.001.  The  rheostat  arm,  R}  can 
be  varied  by  one  ohm  steps  from  0  to  11110  ohms.  This  gives 
a  range  of  measurement  of  unknown  resistances  from  0.001 
ohm  to  11,110,000  ohms.  In  using  such  a  bridge  it  is  best  to 
first  set  the  ratio  arms  equal,  say  1000  ohms  each,  and  obtain 
an  approximate  value  of  the  unknown  resistance.  Then 
change  the  ratio  to  such  a  value  that  R  may  be  given  in  four 


THE  WHEATSTONE  BRIDGE 


67 


figures.     This  will  give  the  resistance  of  the  unknown  to  four 
significant  figures  also. 

A  more  convenient  form  is  the  decade  bridge.  The  rheostat 
arm  is  arranged  on  the  decade  plan  with  one  plug  for  each 
decade.  The  resistance  in  this  arm  is  indicated  by  the  position 
of  the  plugs,  which  always  remain  in  the  box.  The  ratio  arms 


FIG.  29. — Diagram  of  a  decade  Wheatstone  bridge  box. 

consist  of  a  single  series  of  coils  of  1,  10,  100,  1000,  1000, 
10000,  ohms,  but  any  coil  can  be  used  in  either  arm,  (but  of 
course  the  same  coil  can  not  be  used  in  both  arms  at  the  same 
time).  The  connections  not  visible  are  clearly  indicated 
by  lines  drawn  on  the  top  of  the  box.  The  different  parts 
should  be  carefully  compared  with  the  diagram  of  Fig.  26, 
Article  48,  and  the  points  A,  B,  C,  D,  located  before  attempt- 
ing to  use  the  box.  The  resistance  to  be  measured  is 


68  ELECTRICAL  MEASUREMENTS 

joined  to  the  posts  marked  X,  the  battery  and  the  galva- 
nometer as  shown,  with  a  key  in  each  circuit — preferably  a 
successive  contact  key. 

When  starting  to  obtain  a  balance  the  ratio  arms  are  each 
set  at  1000  ohms  (the  figure  shows  only  one  1000  ohm  coil  at 
a,  but  there  are  two  on  the  box),  and  an  approximate  value  of 
the  resistance  determined.  This  is  done  by  shunting  the  gal- 
vanometer with  the  smallest  shunt  available  and  with  R  set  at 
one  ohm  the  keys  are  quickly  tapped  and  the  direction  of  the 
deflection  noted.  The  key  should  not  be  held  down  long 
enough  to  cause  a  large  deflection  as  the  direction  can  be  seen 
from  a  small  one  just  as  well  and  with  less  danger  to  the  gal- 
vanometer. Next,  R  is  set  at  9000  ohms  and  the  key  tapped. 
Usually  this  deflection  will  be  in  the  opposite  direction.  If 
it  is  not  try  zero  and  infinity.  Knowing  that  the  value  of 
X  lies  between  1  and  9000,  say,  this  range  is  divided  by  next 
trying  100  ohms,  and  if  X  is  less  than  this  try  10  ohms.  Sup- 
pose X  is  between  10  and  100.  This  range  is  divided  by  trying 
50,  and  so  on  until  it  is  reduced  to  one  ohm,  say  X  is  found  to  be 
between  68  and  69  ohms.  Then  the  ratio  arms  are  changed  so 
as  to  make  R  come  6800  or  6900.  The  exact  value  for  a  balance 
is  determined  by  continuing  the  same  process  and  is  found,  say, 
when  R  is  6874  ohms.  This  example  then  gives  X  =  68.74 
ohm's. 

If  the  best  balance,  obtained  with  no  shunt  on  the  galvanom- 
eter still  gives  some  deflection,  the  next  figure  for  X  can  be 
obtained  by  interpolation,  but  this  is  not  usually  required.  If 
greater  accuracy  is  desired  it  is  necessary  to  make  a  second 
measurement  with  the  battery  current  reversed  through  the 
bridge.  This  will  reverse  some  of  the  errors,  and  especially 
the  effect  of  thermal  currents  in  the  galvanometer  branch.  The 
mean  of  these  two  measurements  will  then  be  nearer  the  true 
value  of  X  than  either  one  alone. 

Measure  the  resistance  of  several  coils  and  check  the  results 
by  measuring  their  resistance  when  joined  in  series  and 
parallel.  If  some  of  these  coils  have  an  iron  core  notice  the 


THE  WHEATSTONE  BRIDGE  69 

effect  of  first  closing  the  galvanometer  key  and  then  closing  the 
battery  key.     Remember  that  the  formula  for  this  method  was 
deduced  on  the  assumption  that  all  of  the  currents  were  steady, 
and  that  there  was  no  current  through  the  galvanometer. 
The  data  can  be  recorded  as  follows: 


Object  measured 

P 

Q 

R 

X 

53.  Location  of  Faults. — By  a  fault  on  a  telephone,  electric 
light,  or  other  line,  is  meant  any  trouble  by  which  the  insula- 
tion of  the  line  is  impaired,  or  which  interferes  with  the  proper 
working  of  the  line.  The  principal  kinds  of  faults  are  named 
as  follows: 

A  ground  is  an  electrical  connection  more  or  less  completed 
between  the  line  and  the  earth.  In  the  case  of  a  cable  any 
connection  from  one  of  the  wires  to  the  lead  covering  of  the 
cable  constitutes  a  ground. 

A  cross  is  an  electrical  connection  between  two  wires. 

An  open  is  a  break  in  the  line. 

In  testing  for  a  fault  it  is  first  necessary  to  determine  to 
which  of  these  classes  the  given  trouble  belongs.  A  testing 
circuit  is  made  by  connecting  a  battery  in  series  with  a  volt- 
meter or  other  current  indicator.  The  faulty  line  is  then  picked 
out  from  among  the  good  ones  by  one  of  the  methods  outlined 
below. 

Test  for  Grounds. — The  test  to  find  grounded  wires  can  be 
made  at  any  point  along  the  line.  The  battery  side  of  the  test- 
ing circuit  just  described  is  connected  to  the  ground,  and  the 
wire  from  the  voltmeter  side  is  brought  into  contact  successively 
with  each  wire  to  be  examined.  When  the  grounded  wire  is 
reached  the  battery  circuit  is  completed  through  the  earth,  and 
this  will  be  indicated  by  the  deflection  of  the  voltmeter. 


70  ELECTRICAL  MEASUREMENTS 

Tests  of  this  kind  made  with  alternating  current  are  often 
unreliable  because  of  the  capacity  of  the  line. 

Test  for  Opens. — Before  testing  for  opens  the  distant  ends 
of  the  wires  are  joined  together  and  grounded.  The  test  is 
applied  at  the  near  end,  using  the  testing  circuit  in  the  manner 
just  described.  As  each  wire  is  tried  the  voltmeter  will  indi- 
cate the  ground  which  has  been  placed  on  the  other  end,  unless 


FIG.  30. — Picking  out  the  grounded  wire. 

the  line  is  open  at  some  intermediate  point.  If  the  line  is 
broken  the  needle  of  the  voltmeter  will  remain  at  rest,  show- 
ing that  there  is  no  electrical  connection  through  that  wire. 

Test  for  Crosses. — In  testing  for  crosses  the  near  ends  of 
all  the  wires  are  connected  together  and  to  one  end  of  the  test- 
ing circuit.  The  wires  are  then  disconnected  one  at  a  time, 
and  the  free  end  joined  to  the  other  end  of  the  testing  circuit. 


/777Z7 

FIG.  31. — Picking  out  the  broken  line. 

A  movement  of  the  voltmeter  needle  shows  that  the  wire  being 
tested  is  crossed  with  one  of  the  other  wires.  If  there  is  no 
indication  of  a  cross  the  wire  is  laid  at  one  side  and  the  test 
continued  with  other  wires  until  all  the  good  wires  have  been 
removed,  and  only  crossed  wires  remain. 

In  case  there  are  several  sets  of  crossed  wires  among  those 
remaining,  the  near  ends  of  all  the  crossed  wires  should  be 


THE  WHEATSTONE  BRIDGE  71 

separated.  Then  connecting  them,  one  at  a  time,  to  the 
battery  side  of  the  testing  circuit,  touch  each  of  the  other  wires 
in  succession  with  the  wire  from  the  voltmeter.  A  deflec- 
tion indicates  that  the  line  touched  is  crossed  with  the  one 
joined  to  the  battery. 


FIG.  32. — Picking  out  the  crossed  lines. 

Of  course  the  wires  must  not  be  connected  at  the  other  end, 
or  through  any  switchboard  as  this  would  give  the  test  for  a 
cross. 

54.  Methods  for  Locating  Faults. — The  usual  methods  for 
locating  faults  in  a  telephone  line  or  cable  are  based  on  the 
principles '  of  the  simple  slide  wire  bridge.     The   following 
methods  are  illustrative  examples  of  this  kind   of  measure- 
ment and  show  the  general  mode  of  procedure.     They   are 
called  loop  tests  because  the  wire  being  tested  is  joined  to  a 
good  wire  thus  forming  a  long  loop  out  on  one  wire  and  back 
on  the  other. 

In  case  the  line  is  open  so  it  can  not  be  used  as  one  arm  of 
a  Wheatstone  bridge,  the  position  of  the  break  can  be  located 
by  comparing  the  capacity  of  the  line  out  to  the  break  with  the 
capacity  of  similar  lines  whose  length  is  known. 

55.  The  Murray  Loop. — In  the  Murray  Loop  method  the 
grounded  wire  is  joined  to  one  end  of  a  slide  wire  bridge,  as 
shown  at  A,  Fig.  33.     A  second  wire  of  the  same  length  and 
resistance  and  similar  to  the  first,  but  free  from  faults,  is 
joined  at  D  to  the  other  end  of  the  bridge.     These  two  wires 
are  joined  together  at  the  far  end  of  the  line,  thus  forming 
the  loop.     This  loop  is  divided  into  two  portions  by  the  fault 
at  F,  which  in  the  figure  is  assumed  to  be  a  ground.     These 
two  parts  form  two  arms  of  a  Wheatstone  bridge,  the  other 
two  being  formed  by  the  bridge  wire  ACD.     It  is  usually  best 
to  connect  the  galvanometer  and  battery  in  the  position  in- 


72  ELECTRICAL  MEASUREMENTS 

dicated,  the  battery  connection  at  F  being  made  through  the 
earth. 

Let  C  indicate  the  balance  point.  Let  p  denote  the  resist- 
ance of  1  mm.  of  the  bridge  wire,  and  p'  the  resistance  of 
unit  length  of  the  wire  forming  the  loop.  Then  by  the 
principle  of  the  Wheatstone  bridge  for  a  balance, 

flnri  Wl  =  (c  -  a)  pi' 

dp'I  =  (2L  -  d)  pT 

from  which  d  =  2L  — 

c 

where  d  is  the  distance  to  the  fault  and  L  is  the  length  of  the 
faulty  wire. 


niniii  •-* 


FIG.  33. — Locating  the  position  of  the  ground,  F. 

This  determination  can  be  checked  by  exchanging  the  good 
and  bad  wires  in  the  arms  of  the  bridge.  This  change  re- 
quires a  slight  modification  in  the  formula  used  in  solving 
ford 

66.  Fisher's  Method. — It  sometimes  happens  that  no  good 
wire  like  the  grounded  one  can  be  obtained.  It  is  still  possible 
to  locate  the  fault  provided  only  that  two  good  wires  can  be 
obtained,  the  only  requisite  being  that  they  terminate  at  the 
same  point  as  the  faulty  wire. 

First  make  connections  as  in  Fig.  33  using  one  of  the  good 
wires  to  complete  the  loop  with  the  faulty  wire.  Then  as 
before, 

-  -         dpf  (A} 

c  "  Lp'+Hp" 

where  Hp"  is  the  resistance  of  the  good  wire. 


THE  WHEATSTONE  BRIDGE 


73 


Next  use  the  other  good  wire  to  connect  the  battery  to  the 
far  end  of  the  line  as  shown  in  Fig.  34  and  obtain  a  second 
balance  The  ground  at  F  will  make  no  difference  if  there  is 
not  a  second  ground  at  some  other  point  In  this  case,  if  a!  is 
the  reading, 

a'  Lp' 

c      ''    Lp'  +  Hp" 


(B) 


HIM  n 


FIG.  34. — Finding  the  relative  resistance  of  the  two  lines. 

From  (A)  and  (B), 

ad  T  a 

-7  =    T         or        d  =  L—, 


where  as  before,  d  is  the  distance  to  the  fault  and  L  is  the 
length  of  the  faulty  wire. 

57.  Location  of  a  Cross. — The  methods  for  locating  a  cross 
are  similar  to  those  just  given  for  locating  a  ground.     The 


L' 


FIG.  35. — Locating  the  position  of  the  cross,  EF. 

only  difference  is  that  instead  of  Connecting  the  battery  to  F 
by  means  of  the  earth,  as  shown  in  Fig  33,  the  connection  is 


74  ELECTRICAL  MEASUREMENTS 

now  made  by  means  of  the  wire  which  is  crossed  with  the  line 
AB.  The  balance  point  is  found  the  same  as  before,  and  the 
distance  to  the  fault  computed  by  the  formulae  given  above. 
Fig.  35  shows  the  Murray  loop  for  locating  the  cross  EF 
between  the  lines  L  and  H .  Of  course  the  location  of  the  cross 
could  have  been  made  equally  well  by  using  line  H  in  the 
bridge  in  the  place  of  L. 

58.  Location  of  a  Cross.  Wires  unlike. — In  case  the  lines 
L  and  L'  are  unlike  it  will  be  necessary  to  use  another  line  J 
as  shown  in  Fig.  34  and  obtain  a  second  balance  as  in  the  pre- 
ceding method.  The  distance  to  the  cross  is  then  given  by  the 

formula  d  =  L-,  derived  as  before. 

69.  Location  of  Opens. — When  one  of  the  lines  is  broken 
it  will  not  be  possible  to  use  it  for  one  arm  of  a  Wheatstone 


^e 

f 

M 

h 

i 

x^ 

^m                                    L                                n 

FIG.  36. — Finding  the  distance  out  to  the  break  at  6. 

bridge.  If  the  line  is  one  of  a  pair  it  may  have  sufficient 
capacity  to  be  measured.  The  simplest  way  is  to  charge  one 
piece  of  the  broken  wire  and  discharge  it  through  a  ballistic 
galvanometer.  This  deflection  is  compared  with  the  deflec- 
tion obtained  when  a  known  length  of  a  similar  wire  is  charged 
and  discharged  in  the  same  way.  Then 

a     rd' 

d=Ljr, 

Usually  a  more  exact  determination  can  be  made  by  the 
bridge  method  (Article  109)  Let  ac  and  ef  represent  the  two 
wires  of  a  pair,  of  which  ac  is  broken  at  the  point  6.  A  similar 
pair  is  shown  by  ran  and  hj.  The  line  ran  and  the  part  be 


THE  WHEATSTONE  BRIDGE 


75 


of  the  broken  line  are  joined  to  the  non-inductive  resistances 
Rf  and  R"y  as  shown.  The  other  wires  of  these  pairs,  and  the 
remainder  of  the  broken  wire,  are  joined  to  the  ground.  An 
induction  coil,  7,  or  some  other  source  of  alternating  E  M.F., 
is  used  to  charge  the  lines  through  the  resistances.  When 
the  latter  are  adjusted  to  give  a  minimum  sound  in  the 
telephone  T, 

J       1H" 
d=LRr 

60.  Resistance    of   Electrolytes. — When    a    current    flows 
through  an  electrolyte  it  is  accompanied  by  a  decomposition 


FIG.  37. — Resistance  of  an  electrolyte. 

of  the  substance  in  solution.  The  positive  ions  move  in  the 
same  direction  as  the  current,  while  the  negative  ions  travel 
in  the  opposite  direction,  each  being  liberated  at  the  electrodes. 
In  general  this  action  causes  polarization,  which  tends  to 
oppose  the  flow  of  current.  In  order,  therefore,  to  measure 
the  resistance  of  an  electrolyte  it  is  necessary  to  employ  an 
alternating  current.  This  can  be  most  readily  obtained  from 
a  small  induction  coil. 

The  electrolyte  is  placed  in  a  suitable  cell,  and  made  the 
fourth    arm    of   a    Wheatstone    bridge,    the    induction    coil 


76  ELECTRICAL  MEASUREMENTS 

being  used  in  place  of  the  usual  battery.  The  resistance  of 
the  electrolyte  can  then  be  determined  by  the  Wheatstone 
bridge  method  in  the  usual  way,  and  when  the  bridge  is 
balanced, 


Since  an  alternating  current  is  employed,  this  balance  can 
be  found  by  means  of  a  telephone  receiver  connected  in  the 
usual  place  for  a  galvanometer.  For  purposes  of  instruction 
the  best  form  of  cell  for  holding  the  electrolyte  is  a  cylindrical 
tube  with  a  circular  electrode  closing  each  end.  The  resist- 
ance measured  by  the  bridge  is  then  the  resistance  of  the 
electrolyte  between  the  two  electrodes,  and  knowing  the  resist- 
ance of  this  column  of  the  electrolyte  the  resistivity,  s,  of  the  so- 
lution can  be  calculated  the  same  as  for  metallic  conductors,  or, 

A 


Where  A  is  the  cross  section  of  the  tube  containing  the  solution 
and  L  is  the  distance  between  the  electrodes. 

The  conductivity  of  the  solution,  c,  is  the  reciprocal  of 
this,  or, 

1         L 

C     =      -      =       —T* 

s         rA 

Since  the  resistance  of  an  electrolyte,  or  more  strictly,  its 
conductivity,  depends  upon  the  amount  of  the  substance  in 
solution  —  that  is,  upon  the  number  of  ions  per  cubic  centi- 
meter —  if  we  wish  to  compare  the  conductivities  of  different 
electrolytes  it  is  necessary  to  express  the  concentrations  in 
terms  of  the  number  of  ions  per  cubic  centimeter.  This  is 
usually  stated  in  terms  of  the  number  of  gram  molecules  of 
substance  that  are  dissolved  in  one  liter  of  the  solution.  For 
the  purpose  of  this  experiment  it  is  necessary  to  express  the 
concentrations  in  terms  of  the  number  of  gram-molecules  in 
1  cc  of  the  solution.  The  molecular  conductivity,  /*,  of 


THE  WHEATSTONE  BRIDGE  77 

an  electrolyte  is  then  defined  as  the  conductivity  per  gram- 
molecule  of  salt  contained  in  each  cubic  centimeter  of  solution. 

L          bL 


m  ~  ms  ~  mrA  ~  amRA 

where  m  =  number  of  gram-molecules  in  1  cc.  of  the  solution. 
The  most  interesting  application  of  the  conductivity  of 
solutions  is  the  knowledge  it  gives  regarding  the  degree  of 
dissociation  of  the  dissolved  substance.  The  conductivity  of 
an  electrolyte  is  due  entirely  to  the  ions  it  contains  and  is 
directly  proportional  to  the  number  of  ions  per  cubic  centi- 
meter. Most  salts  are  completely  dissociated  in  very  dilute 
solutions,  and  therefore  the  molecular  conductivity  of  such 
solutions  is  not  increased  by  further  dilution.  Call  this  value 
/i0.  Then  if  /*  denotes  the  molecular  conductivity  of  a  more 
concentrated  solution  of  the  same  salt,  the  relative  dissociation 
in  this  solution  is, 

M 

OL    —    —  * 

Mo 


Express  results  by  means  of  a  curve,  using  values  of  /*  for  ordi- 
nates  and  the  corresponding  values  of  —  (  =  number  of  cubic 
centimeters  containing  1  gram  molecular)  as  abscissae. 


CHAPTER  V 
THE  WHEATSTONE  BRIDGE  (Continued) 

61.  The  Slide  Wire  Bridge  with  Extensions.— The  measure- 
ment of  resistances  by  the  slide  wire  bridge  can  be  made  with 
more  precision  by  using  a  longer  bridge  wire.  The  un- 
certainty in  locating  the  balance  points  probably  will  be  about 
the  same,  but  since  the  distance,  a'  —  a",  between  the  two 
balance  points  is  increased,  the  percentage  error  will  be  less. 

As  it  would  be  inconvenient  to  have  the  apparatus  much 
over  a  meter  in  length,  and  as  only  the  middle  portion  of  the 


FIG.  38. — Bridge  with  extensions,  m'  and  m". 

bridge  wire  is  used  in  making  careful  measurements,  the 
effective  length  of  the  bridge  wire  is  increased  by  adding  a 
resistance  at  each  end.  These  extensions  may  consist  of  know  n 
lengths  of  wire  similar  to  that  used  for  the  bridge  wire — or 
any  two  equal  resistances  may  be  used  and  their  equivalent 
lengths  determined  experimentally  by  the  method  shown  below. 
The  meter  of  wire  provided  with  a  scale  then  becomes  only  a 
short  portion  along  the  middle  of  the  total  length  of  the 
bridge  wire.  While  this  arrangement  makes  possible  a 
greater  precision  of  measurement,  it  also  lessens  the  range  of 

78 


THE  WHEATSTONE  BRIDGE  79 

the  bridge,  as  only  those  balances  which  fall  on  this  limited 
section  of  the  wire  can  be  read. 

The  extensions  are  placed  in  the  outside  openings  on  the 
back  of  the  bridge,  between  the  ends  of  the  bridge  wire  and 
the  battery  connections.  They  should  be  nearly  equal. 
Let  ra'  and  m"  denote  the  number  of  millimeters  of  bridge 
wire  having  the  same  resistances  as  each  extension,  respectively, 
and  let  L  denote  the  total  length  of  the  bridge  wire  including 
both  of  its  extensions. 

With  the  resistance,  x,  to  be  measured,  and  the  known 
resistance,  R,  in  the  middle  openings  of  the  bridge,  as  shown 
in  Fig.  38,  the  first  balance  point  is  found.  The  reading  on 
the  scale  at  this  point  will  be  called  a'.  Then 

jc_  mr  +  a'  m'  -f-  a' 

R~  m"  +  V  ~  L  -  (ro'  +  a') 

Exchanging  the  positions  of  x  and  R}  and  calling  the  scale 
reading  at  the  new  balance  point  a", 


R  m'  +  a" 

And  by  the  additions  of  proportions, 

*  _  L  +  (ar  -  a")  _  L  +  d 
R-  L  -(a'  -  a")  ~  L  -d 


(3) 


It  is  evident  that  this  arrangement  reduces  the  range  of  the 
bridge,  for  only  those  values  of  x  can  be  measured  which  are 
near  enough  equal  to  R  to  give  balance  points  on  the  scale. 
But  what  is  lost  in  range  is  more  than  made  up  in  the  greater 
precision  of  measurement. 

Dividing  out  the  fraction  in  (3)  gives 


80  ELECTRICAL  MEASUREMENTS 

All  the  terms  after  the  second  are  negligible  if  d  is  small  in 
comparison  with  L,  so  that, 

x  =  R  +  2R~  (5) 

The  only  part  of  x,  then,  which  is  measured  by  the  bridge  is  the 
second  term,  and  a  small  error  in  it  will  only  slightly  affect  the 
computed  value  of  x. 

62.  To  Find  the  Length  of  the  Bridge  Wire  with  Its  Ex- 
tensions.— The  total  length  of  the  bridge  wire,  including  the 
extensions  at  each  end,  can  be  determined  as  follows.  A  good 
resistance  box,  P,  is  used  in  place  of  the  unknown  resistance  x 
shown  in  the  figure  of  the  preceding  section.  Then  with  both 
extensions  connected  in  the  bridge,  the  values  of  P  and  R  are 
adjusted  to  bring  the  balance  point  near  one  end  of  the  scale. 
Let  a'  denote  this  scale  reading,  corrected  if  necessary  by  the 
calibration  curve  for  this  wire  when  used  as  a  simple  bridge. 
Then 

P  m'  +  a'  m'  +  a' 

P  -h  R  ''  ~'  m'  +  c  +  m"  L 

where  c  denotes  the  original  length  and  L  the  total  length  of  the 
bridge  wire. 

Exchanging  P  and  R,  the  balance  falls  near  the  other  end  of 
the  wire  and 

_R  m'  +  a" 

P 
By  subtraction, 


whence 

L  = 


R  - 


This  method  may  also  be  used  to  determine  whether  there  is 
any  extra  resistance  in  the  straps  and  connections  at  the  ends  of 
the  usual  meter  of  bridge  wire. 


THE  WHEATSTONE  BRIDGE  81 

Problem.  —  In  calibrating  a  bridge  wire  it  was  found  that  for  P  = 
100,  and  Q  =  900  ohms,  the  balance  point  fell  at  95  on  the  scale; 
while  for  P  =  900  and  Q  =  100  the  balance  point  was  at  903.  What 
is  the  effective  length  of  the  bridge  wire?  Ans.  L  =  1010  mm. 

63.  To  Calibrate  the  Slide  Wire  Bridge  with  Extensions.— 
The  formula  deduced  for  this  method  in  eq.  5  above  works 
very  well  as  long  as  x  and  R  are  nearly  equal;  but  several 
errors  may  occur  in  its  use,  the  principal  of  which  are: 

1.  Using  a  wrong  value  for  L. 

2.  Neglecting  all  the  terms  containing  d  in  powers  higher 
than  the  first. 

3.  Errors  in  the  determination  of  d,  due  to  non-uniform 
bridge  wire,  scale  errors,  etc. 

The  method  of  calibration  described  below  corrects  for  all  of 
these  errors  at  once  by  finding  a  correction  to  be  added  to  the 

observed  value  of  d,  which  will  give  to  (1  +  2j\  the  true 

value  of  -5- 
rt 

With  the  bridge  set  up  as  shown  in  Fig.  38,  with  the  exten- 
sions in  place,  and  two  good  resistance  boxes,  P  and  Q  in  place 
of  x  and  R,  we  have, 


and  solving  for  d  gives, 

d  =  ^(P-Q)  (B) 

This  is  the  value  that  d  must  have  in  order  that  (A)  shall 
give  the  correct  values  of  the  resistances  being  measured. 

Starting  with  P  and  Q  each  1000  ohms,  the  value  of  d  should 
be  zero.  Then  increasing  P  by  successive  small  steps  the 
corresponding  observed  values  of  d  can  be  determined.  These 
observed  values  of  d  will  not  agree  with  those  computed  from 
(B)  above,  and  therefore  if  used  in  (A)  will  not  give  the  correct 
values  for  P.  This  is  because  of  the  errors  noted  above.  It 


82 


ELECTRICAL  MEASUREMENTS 


is  therefore  necessary  to  add  to  the  observed  length  of  bridge 
wire,  d,  a  certain  amount,  c,  such  that 


and  this  corrected  value,  ef  ,  is  what  should  be  used  in  eq.  5 
above. 

In  the  present  case  where  P  and  Q  are  known,  the  values  of 
d'  are  computed  from  eq.  (B),  while  the  corresponding  values 
of  d  are  observed  on  the  bridge  wire.  The  differences  give  the 
values  of  c,  and  a  calibration  curve  can  be  drawn,  as  shown, 
which  will  give  the  correction  to  be  used  at  each  point.  The 
correction  increases  rapidly  with  d  owing  to  t  he  increasing  im- 
portance of  the  second  error  noted  above. 

In  case  d  is  not  zero  when  P  and  Q  are  nominally  equal  it 
means  that  one  is  really  a  little  larger 
than  the  other.  Let  d0  denote  this 
value  of  d.  This  can  be  reduced  to  zero 
by  adjusting  the  value  of  Q;  or  if  it  is 
more  convenient,  d0  may  be  subtracted 
from  each  value  of  d  throughout  this 
calibration,  as  a  sort  of  zero  correction. 
The  data  may  be  recorded  as  below. 


FIG.  39. — Calibration 
curve. 


To  CALIBRATE  BRIDGE  No.  .      .  WITH  EXTENSIONS  No AND  No.  . . . 


P 

Q 

a' 

a" 

d 
=  a'-a" 

d' 
Eq.  B 

c 
=  d'-d 

64.  Advantages  of  the  Double  Method. — Wheri  it  is  desired 
to  use  the  slide  wire  bridge  with  some  degree  of  precision  several 
precautions  are  necessary  in  order  to  avoid  the  principal  errors. 
Prominent  among  these  are  the  effects  due  to  thermal  elec- 


THE  WHEATSTONE  BRIDGE  83 

tromotive  forces  in  the  galvanometer  circuit,  to  balance  which 
it  is  necessary  to  set  the  sliding  contact  on  the  wire  at  a  point 
somewhat  to  one  side  of  the  true  balance  point  to  obtain  zero 
deflection  of  the  galvanometer.  If  the  scale  is  displaced 
endwise  with  respect  to  the  wire,  or  if  the  index  from  which 
the  readings  are  taken  is  not  exactly  in  line  with  the  point  at 
which  contact  is  made  on  the  wire,  the  effect  is  much  the  same. 
Let  a'  denote  the  observed  reading  on  the  scale,  and  a'  +  / 
the  true  balance  point,  where/  denotes  the  displacement  of  the 
reading  due  to  the  causes  noted  above.  The  actual  value  of  / 
is  unknown  but  it  is  constant  in  amount  and  sign,  at  least  while 
one  set  of  readings  is  being  taken.  For  a  balance  of  the  bridge 
with  x  and  R  in  the  positions  shown  in  Fig.  27,  we  have 

x  a'  +f 

R  "  c  -  (a'+f) 

where  c  is  the  total  length  of  the  bridge  wire  expressed  in  the 
same  units  as  a!  and  / — usually  in  millimeters. 

Exchanging  x  and  R  gives  a  new  balance  at  a",  and 

x  '      c  -  (a"  +  /) 
R  '          a"  +  / 

Combining  these  two  expressions  by  the  addition  of  proportions 
gives 

x        c  +  (a'  -  a")        c  +  d 
R  '-  =  c  -  (a'  -  a")   ''~~  c  -  d 

It  is  seen  that  /  has  been  eliminated  by  this  double  method 
and  the  only  measured  quantity  appearing  in  the  final  expres- 
sion is  d,  the  length  of  the  wire  between  the  two  observed  balance 
points.  The  value  of  c  should  be  determined  by  the  method 
given  in  Article  62  as  this  may  be  greater  than  the  meter  of 
bridge  wire  because  of  the  added  resistance  of  the  copper  straps 
and  the  connections  at  each  end  of  the  wire.  The  "bridge 
wire"  really  includes  all  of  the  resistance  from  A  to  D.  How- 
ever, if  d  is  small  a  slight  uncertainty  in  the  value  of  c  will  pro- 


84  ELECTRICAL  MEASUREMENTS 

duce  a  negligible  error  in  the  computed  value  of  x.  This 
means  that  R  should  be  taken  as  near  to  the  value  of  x  as  is 
convenient. 

65.  The  Best  Position  of  Balance,  (a)  Simple  Bridge.  —  The 
formula  deduced  in  Article  49  for  the  value  of  a  resistance 
measured  by  the  simple  slide  wire  bridge  is 


Suppose  that  the  value  of  a  that  is  read  on  the  scale  and  put  into 
this  formula  is  too  large  by  F0  mm.  This  will  make  the  com- 
puted value  of  x  too  large  by  Fx,  say.  It  is  required  to  find  the 
quantity  Fx.  Let  the  relationship  between  Fx  and  Fa  be 
denoted  by  m. 
then  Fx  =  m  Fa  .  (B) 

and  it  remains  to  find  m. 

•p 

From  B,  m  =  ~r* 

fa 

Fa  is  a  small  part  of  a,  and  Fx  is  the  corresponding  small  part 
of  x.     In  the  notation  of  the  calculus  this  would  be  written, 


dx 

da 

and  if  Fa  is  small,  m'  —  m,  whence  B  becomes 


, 
m   =  da 


From  A, 

dx  _       (c  —  a)  +  a  c 

da  ~  (c  —  a)2  (c  —  a)2 

and 


(D) 


But  the  actual  error  made  in  finding  the  value  of  x  is  not  of  as 
much  importance  as  the  relative  error.  It  is  evident  that  an 
error  of  one  ohm  in  a  total  of  ten  ohms  is  a  very  different  thing 


THE  WHEATSTONE  BRIDGE  85 

from  an  error  of  one  ohm  in  a  thousand  ohms.  The  relative 
error  ig  the  ratio  of  the  actual  error  to  the  total  quantity 
measured.  Thus  from  (A)  and  (D)  the  relative  error,  e,  is 


*        F 

—  =  r 


x         °  (c  -  d)a 

From  this  it  appears  that  even  the  relative  error  is  not  the 
same  for  the  same  error  in  reading,  but  it  depends  upon  the 
value  of  a.  Examining  this  expression  for  a  minimum  value 
of  e 

de_  _    -  c(c  -  2a)       _ 

da~   ((c  -a)a)**a~  °* 

This  is  satisfied  if  (c  -  2a)  =  0. 

Thus  in  reading  the  value  of  o,  a  given  error  (say  1  mm.) 
will  produce  the  least  effect  on  the  computed  value  of  x  when 
the  balance  point  comes  at  the  middle  of  the  bridge  wire. 

(6)  Double  Method 

The  above  discussion  applies  to  the  simple  slide  wire  bridge. 
When  the  double  method  is  used  the  formula  for  the  resistance 
being  measured  is 

1000  +  (a'  -  a") 
*  1000  -  (a'  -  a") 
as  deduced  in  Article  51. 
Writing  this  as 


it  can  be  shown  in  the  same  way  that  a  given  error  in  measuring 
z  will  have  the  least  effect  upon  the  computed  value  of  x  when 
z  =  0,  that  is,  when  both  a'  and  a"  are  at  the  middle  of  the 
bridge  wire.  In  this  case 

dx  2c 

dz  ~    K(c  -  z)2 
and 

2c  2cF, 


e  = 


\c  -z)(c+z)       c2  - 
which  evidently  is  a  minimum  when  z  =  0. 


86  ELECTRICAL  MEASUREMENTS 

66.  Sources  of  Error  in  Using  the  Slide  Wire  Bridge.— 
These  may  be  summarized  as 

I.  Errors  in  setting,  due  to 

a.  Thermal  currents. 

b.  Contact  maker  not  in  line  with  index. 

c.  Non-uniform  wire  or  scale. 

d.  Ends  of  wire  and  scale  not  coincident. 
II.  Errors  in  reading. 

e.  The  position  of  balance. 

/.   True  value  of  R,  loose  plugs,  etc. 

The  effect  of  a  and  b  can  be  eliminated  by  using  the  double 
method,  as  explained  in  Article  64. 

The  only  way  to  avoid  the  effect  of  c  or  d  is  by  calibration 
of  the  bridge  wire  and  correcting  all  readings. 

The  error  in  reading  the  position  of  the  index  after  a  balance 
has  been  found  is  often  greater  than  the  uncertainty  of  the 
setting.  In  the  preceding  section  it  was  shown  that  this 
error,  which  is  about  the  same  for  all  parts  of  the  scale,  has  the 
least  effect  on  the  computed  value  of  x  when  the  reading  is  near 
the  middle  of  the  bridge  wire. 

The  error  in  the  resistance  coils  of  a  good  box  is  very  small. 
However,  the  value  of  R  read  from  the  box  and  used  in  com- 
putations may  be  very  different  from  the  actual  resistance  of 
the  experiment.  If  some  of  the  plugs  are  loose,,  or  make  poor 
contact  because  of  dirt  or  corrosion,  the  resistance  may  be 
considerably  increased.  Moreover,  the  resistance  actually 
used  in  the  bridge  includes  all  the  connections  and  lead  wires 
used  to  join  the  box  to  the  bridge.  In  the  same  way  the  re- 
sistance measured  includes  the  lead  wires  and  connections. 

67.  The  Direct  Reading  Bridge. — The  ordinary  slide  wire 
bridge  used  with  extension  at  each  end  of  the  bridge  wire  can 
easily  be  made  into  a  direct  reading  bridge.     Its  peculiarity 
lies  in  its  calibration  and  the  adjustment  of  the  extensions. 
When  a  large  number  of  resistances  are  to  be  measured  the 
double  readings  and  consequent  calculations  are  too  slow  a 
method.     It  may  then  be  better  to  adjust  the  extensions  to 


THE  WHEATSTONE  BRIDGE  87 

make  the  total  length  of  the  bridge  wire  40  meters,  or  each 
extension  equivalent  to  19.5  meters  of  the  bridge  wire.  With 
this  length,  and  adjusted  to  bring  the  middle  point  at  50  on 
the  scale,  the  balance  point  will  fall  at  51  for  values  of  x  larger 
than  R  by  0.1  per  cent.,  and  1  cm.  further  from  the 
center  for  each  additional  0. 1  per  cent,  between  x  and  R.  Only 
a  single  balance  is  obtained  and  it  is  then  very  easy  to  read 
from  the  scale  the  percentage  by  which  x  is  larger  than  R. 
This  relation,  which  is  exact  at  the  center,  departs  more  and 
more  from  the  scale  reading  as  the  balance  point  falls  further 
from  the  center.  The  slight  corrections  needed  are  readily 
obtained  from  the  calibration  curve. 

For  the  calibration  of  the  bridge  two  well  adjusted  resistance 
boxes  are  inserted  in  the  back  openings  of  the  bridge.  By 
setting  each  box  at  5000  ohms  the  middle  point  of  the  total 
bridge  wire,  including  the  extensions,  is  located.  One  or  both 
of  the  extensions  should  be  adjusted  to  bring  this  point  to  50 
on  the  scale.  If  now  one  of  the  boxes,  say  Q,  is  increased  by 
5  ohms,  a  new  balance  point  is  obtained;  and  this  point 
corresponds  to  values  of  Q  which  are  0.1  per  cent,  larger  than 
P.  Q  is  increased  by  5  ohm  steps,  giving  balance  points 
corresponding  to  0.2  per  cent.,  0.3  per  cent.,  etc.  These 
results  may  be  plotted  as  a  calibration  curve,  but  it  is  more 
legible  to  plot  only  the  differences  between  the  correct  readings 
and  those  actually  read  from  the  scale.  The  battery  should  be 
reversed  and  the  mean  of  the  two  readings  employed  in  plotting  . 
the  curve. 

In  using  the  bridge  to  measure  an  unknown  resistance  a 
single  balance  point  is  obtained  (or  the  mean  of  two  if  reversing 
the  battery  makes  any  change).  This  reading  gives  at  once 
the  approximate  percentage  by  which  x  is  larger  than  R.  A 
glance  at  the  calibration  curve  gives  the  correction  to  be  added 
to  obtain  the  correct  percentage.  A  short  multiplication 
gives  the  results  in  ohms. 

68.  Measurement  of  Resistance  by  Carey  Foster's  Method. 
— One  of  the  most  exact  methods  for  comparing  two  resist- 


88 


ELECTRICAL  MEASUREMENTS 


ances  is  the  one  devised  by  Prof.  Carey  Foster  of  England. 
The  Wheatstone  bridge  is  arranged  in  the  same  manner  as 
was  used  for  the  slide-wire  bridge  with  extensions,  except  that 
now  the  extensions  become  the  resistances  to  be  compared. 

Thus  in  the  figure  let  S  be  the  resistance  that  is  to  be  com- 
pared with  R.  These  two  are  placed  in  the  bridge  as  shown, 
being  connected  together  by  the  bridge  wire.  The  other  arms 
of  the  bridge,  P  and  Q,  become  merely  ratio  coils  and  may 
have  any  value  although  they  must  be  nearly  equal  to  each 
other  in  order  that  the  balance  may  fall  on  the  bridge  wire, 


FIG.  40. — Carey  Foster  bridge,  comparing  R  and  S. 

and  for  the  most  sensitive  arrangement  all  the  arms  of  a 
Wheatstone  bridge  should  be  nearly  equal.  For  the  greatest 
accuracy  R  and  S  should  each  be  in  a  constant  temperature 
oil  bath. 

Let  the  balance  point  be  found  by  moving  the  contact  C 
along  the  wire  until  a  point  is  reached  for  which  there  is  no 
deflection  of  the  galvanometer.  Let  a'  be  the  scale  reading 
at  this  point.  It  is  immaterial  whether  this  scale  extends  the 
entire  length  of  the  bridge  wire  or  not. 

Now  let  R  and  S  be  exchanged  with  each  other.  This  will 
make  no  difference  in  the  total  length  of  the  extended  bridge 
wire,  ACD.  But  if  the  resistance  of  S  is  less  than  that  of  R 
the  new  balance  point  will  not  fall  at  the  same  point  as  before, 
for  it  will  be  necessary  to  add  to  S  enough  of  the  bridge  wire  to 
make  the  part  AC  the  same  as  before.  The  resistance  of  AC 
in  the  first  instance  is 


THE  WHEATSTONE  BRIDGE  89 

After  R  and  S  have  been  exchanged  the  resistance  of  AC'  is 


j-  ~r  v 

where  T  denotes  the  total  resistance  of  A  CD.     Equating  these 
two  expressions 

R  +  a'p  =  S  +  a"p, 
and 

S  =  R  -  (a"  -  a')p 

where  p  is  the  resistance  per  unit  length  of  the  bridge  wire. 

69.  To  Determine  the  Value  of  p. — The  resistance  per  unit 
length  of  bridge  wire  can  be  readily  determined  by  the  same 
arrangement  as  shown  above.  Let  R  and  S  be  two  resistances 
differing  by  a  small,  but  very  accurately  known,  difference. 
For  example,  R  =  0  and  S  =  0.500  ohm  could  be  used  if 
these  values  are  definitely  known.  Or,  R  =  1.0  ohm  and  S 
composed  of  a  one  ohm  coil  in  parallel  with  a  ten  ohm  coil 
would  give  a  smaller  difference  which  could  be  used,  provided 
again  the  values  of  the  original  coils  are  really  known.  It 
is  not  essential  that  the  resistances  of  the  coils  be  precisely 
one  ohm,  or  ten  ohms,  but  it  is  necessary  that  the  actual  values 
be  known  in  order  that  the  small  difference  between  them  can 
be  accurately  computed. 

Then  since  R  -  S  =  (a"  -  a'}p 

it  follows  that 

R  -S 


70.  Temperature  Coefficient  of  Resistance.  —  An  interesting 
application  of  the  above  methods  for  the  accurate  measure- 
ment of  resistance  is  found  in  the  determination  of  the  tem- 
perature coefficient  of  resistance  for  various  metals  and  alloys. 
When  the  temperature  of  a  conductor  is  varied  its  resistance 
generally  changes  also,  the  usual  relation  being  that  an  increase 
of  temperature  is  accompanied  by  an  increase  in  resistance. 
The  amount  of  this  increase  will  evidently  be  greater  for  a 
large  resistance  than  for  a  small  one,  and  it  is  conveniently 


90  ELECTRICAL  MEASUREMENTS 

expressed  as  a  certain  fraction  of  the  original  resistance. 
Thus  in  symbols 

dR 

~df  =  atR<  (!) 

where  Rt  is  the  resistance  of  the  conductor  at  some  one 
definite  temperature,  and  the  coefficient  at  denotes  the  frac- 
tion by  which  each  ohm  of  Rt  changes  per  degree  change  in 
temperature. 

If  the  change  in  resistance  per  degree  is  constant  over  a 
considerable  range  of  temperature,  then 


and 

RI  —  RZ     1 

at  =  —.  -  -.  —  ~r 
h  —  tz      Kt 

It  should  be  noted  that  this  value  of  a  applies  to  the  whole 
temperature  range,  ti  —  tz,  but  its  numerical  value  depends 
upon  the  value,  Rt,  that  is  taken  as  the  true  resistance  of  the 
coil  under  consideration. 

70A.  Determination  of  the  Temperature  Coefficient.  —  In 
this  experiment  five  coils  of  different  metals  are  arranged  in 
an  oil  bath  where  the  temperature  can  be  raised  as  desired. 
A  very  convenient  arrangement  is  to  use  an  electric  water 
heater  to  hold  the  oil  bath.  With  a  suitable  resistance  in 
series  this  can  be  easily  warmed  and  maintained  at  the  desired 
temperatures.  Starting  at  room  temperature  the  resistance 
of  each  coil  is  carefully  determined.  A  Wheatstone  bridge 
box  gives  a  convenient  and  accurate  method  for  resistances 
having  large  temperature  coefficients  like  copper  and  iron. 
For  alloys  like  German  silver  and  manganin  it  is  better  to  use 
a  more  delicate  method,  such  as  the  Carey  Foster  bridge. 

After  the  resistances  of  the  coils  have  been  obtained  at 
room  temperature  the  bath  is  warmed  ten  or  fifteen  degrees, 
and  when  things  have  become  steady  at  the  new  temperature 
the  resistances  are  again  measured.  In  the  same  way  the 


THE  WHEATSTONE  BRIDGE  91 

resistances  are  determined  at  five  or  six  different  temperatures, 
and  the  results  plotted  as  a  temperature-resistance  curve  for 
each  coil.  The  slope  of  this  curve  is  dR/dt  of  Eq.  (1).  The 
resistance  of  the  coil  at  20°  C.  can  be  read  from  the  curve. 
Then 


where  the  values  of  R\  and  Rz  are  obtained  from  the  curve  at 
points  corresponding  to  the  temperatures  ti  and  tz.  Ex- 
terpolating  the  curve  back  to  0°  C.,  the  value  of  RQ  can  be 
obtained  from  the  curve  in  the  same  way,  and, 

1       RI  —  RZ 
a°  ~  ^~     ~~  ~ 


70B.  Use  of    the  Temperature    Coefficient.—  If   20°  C.  is 

taken  as  the  standard  temperature  then 

RI  —  RZ 

Rzofyi  —  tz) 

In  using  this  formula  for  finding  the  resistance  at  any  desired 
temperature  t\,  we  have 

R,  =  R2  +  a20R20(ti  -  fe). 
and  further,  if  t2  is  also  20°  C. 

Ri  =  #20(1  +  «2o(«i  -  20)) 
If  0°  C.  is  taken  as  the  standard  temperature, 
Ri  = 


CHAPTER  VI 
MEASUREMENT  OF  CURRENT 

71.  Hot  Wire  Ammeter. — When  an  electric  current  flows 
through  a  wire  one  of  the  most  noticeable  effects  is  that  the  wire 
becomes  warmed.     The  direct  result  of  this  is  an  increase  in 
the  length  of  the  wire.     It  thus  becomes  possible  to  measure 
the  value  of  the  current  in  terms  of  the  change  in  length  of 
such  a  wire,  and  some  ammeters   are   constructed   on   this 
principle. 

72.  The  Weston  Ammeter. — In  addition  to  heating  the  wire 
the  current  produces  a  magnetic  effect  in  the  surrounding 
space.     This  is  manifest  by  its  action  upon  a  magnetic  needle 
near  it,  or  by  the  force  which  the  wire  itself  experiences  when 
in  the  magnetic  field  of  a  magnet  or  another  current.     This 
effect  furnishes  the  basis  for  defining  the  value  of  unit  current 
(see  Introduction,  Article  4)  and  the  precise  relation  between 
the  current  and  the  force  acting  upon  the  wire  is  fully  worked 
out  in  Chapter  XI.     From  this  it  follows  that  a  loop  of  wire 
carrying  a  current  in  a  magnetic  field  will  tend  to  turn  in  a 
definite  direction  according  to  the  direction  of  the  current. 

In  the  Weston  ammeter  the  current  passes  through  a  coil 
of  many  turns  of  fine  wire  wound  on  a  rectangular  form.  This 
coil  is  mounted  on  jeweled  bearings  and  can  turn  in  the  mag- 
netic field  between  the  poles  of  a  strong  permanent  magnet. 
It  is  held  in  a  definite  position  by  a  spiral  spring  at  either  end. 
When  a  current  is  passed  through  the  coil  it  turns  until  the 
torque  of  the  springs  is  sufficient  to  balance  the  couple  due  to 
the  forces  between  the  current  and  the  magnetic  field.  A 
pointer  attached  to  the  coil  moves  over  the  scale  and  indicates 
the  angle  turned  through;  the  scale  is  not  graduated  in  degrees, 

92 


MEASUREMENT  OF  CURRENT  93 

but  in  terms  of  the  current  required  to  produce  the  deflec- 
tion. The  scale  is  thus  direct  reading  and  gives  the  value  of 
the  current  in  amperes. 

In  ammeters  for  measuring  large  currents  a  low  resistance 
shunt  is  placed  in  parallel  with  the  moving  coil  to  allow  only 
a  moderate  current  through  the  latter.  The  scale  is  then 
graduated  to  read  the  value  of  the  large  total  current  through 
both  .the  coil  and  its  shunt.  This  arrangement  is  entirely 
similar,  in  principle,  to  the  voltmeter  and  shunt  described  in 
Article  23,  the  moving  coil  system  acting  as  a  sensitive 
voltmeter. 

73.  The  Weston  Voltmeter. — The  construction  of  a  volt- 
meter is  the  same  as  an  ammeter,    except  that  instead  of 
having  a  shunt  in  parallel  with  the  moving  coil  there  is  a  high 
resistance  in  series  with  the  coil.       The  current  which  will  then 
flow  through  the  instrument  depends  upon  the  E.M.F.  applied 
to  the  terminals  of  this  resistance,  and  the  numbers  written 
on  the  scale  are  not  the  values  of  the  currents  in  the  coil,  but 
are  the  corresponding  values  of  the  fall  of  potential  over  the 
resistance  of  the  voltmeter.     Therefore  it  is  sometimes  said 
that  a  voltmeter  is  really  only  a  sensitive  ammeter  measuring 
the  current  through  a  fixed  high  resistance;  while  an  ammeter 
is  really  only  a  sensitive  voltmeter  measuring  the  fall  of  potential 
over  a  low  resistance. 

74.  Galvanometers. — The  D' Arsonval  type  of  galvanometer 
consists  of  a  moving  coil  suspended  between  the  poles  of  a 
permanent   magnet   something   like   the  arrangement  in   an 
ammeter,  but  greater  sensitiveness  is  secured  by  using  a  long, 
fine  strip  of  phosphor  bronze  for  the  suspension  in  place  of 
the  jewel  bearings.     Such  a  galvanometer  measures  very  small 
currents  and  is  useful  in  measurements  where  the  current  is 
small  or  is  made  zero  in  the  final  adjustment.     It  can  be  used 
to  measure  larger  currents  by  using  a  low  resistance  shunt,  as 
in  the  ammeter;  and  it  will  serve  as  a  voltmeter  when  used  in 
series  with  a  high  resistance. 

Since  the  scale  from  which  is  read  the  deflection  of  the 


94  ELECTRICAL  MEASUREMENTS 

galvanometer  usually  is  divided  into  millimeters,  it  will  be 
necessary  to  calibrate  it  in  order  to  read  the  value  of  the 
current.  A  setup  for  determining  the  figure  of  merit  can  be 
used  (see  Figure  21)  and  deflections  corresponding  to  different 
values  of  the  current  can  be  observed.  A  curve  can  then  be 
plotted  between  deflections  and  currents,  and  from  this  curve 
can  be  read  the  value  of  the  current  corresponding  to  any 
deflection. 

75.  The  Tangent  Galvanometer. — In  the  instruments  here- 
tofore described  it  has  been  supposed  that  the  scale  was  gradu- 
ated to  read  the  value  of  the  current  directly  in  amperes,  or 
that  it  could  be  calibrated  so  to  read.  But  nothing  has  been 
said  as  to  how  the  value  of  a  current  can  be  expressed  in  terms 
of  the  ampere  as  defined  in  the  Introduction.  The  tangent 
galvanometer  furnishes  the  means  for  establishing  the  value 
of  a  currnet  in  terms  of  the  C.G.S.  unit. 
The  method  is  readily  understood  and 
formerly  was  the  principal  method  for  deter- 
mining the  absolute  value  of  a  current. 
Other  methods  (see  Chapter  VII)  now  offer 
more  precise  measurements,  but  the  tangent 
FIG.  41. — Re-  galvanometer  is  as  accurate  as  it  ever  was 

sultantof  two  an(j  it  stm  hoials  sufficient  historical 
magnetic  fields.  .-'  .  ,. 

interest  to  warrant  its  description. 

The  tangent  galvanometer  consists  of  a  coil  of  relatively 
small  cross  section  and  large  diameter.  At  the  center  is  a 
short  magnetic  needle,  suspended  by  a  fine  silk  or  quartz 
fiber  and  carrying  either  a  long  pointer  or  a  mirror  for  use  with  a 
telescope  and  scale.  When  hanging  freely  the  needle  will 
point  north  and  south  in  the  magnetic  msridian.  The  large 
coil  should  stand  vertically  in  this  same  meridian.  When  a 
current  is  passed  through  the  coil  the  magnetic  field  at  the 
center  of  the  coil  due  to  this  current  will  be  directed  east  and 
west.  The  resultant  field  due  to  the  combination  of  this  field 
with  the  original  field  H,  of  the  earth,  will  be  in  a  direction  in- 
termediate between  the  two  and  making  an  angle  0,  say,  with 


MEASUREMENT  OF  CURRENT  95 

the  direction  of  the  latter.  The  needle,  when  free  to  turn,  will 
take  up  this  resultant  direction;  therefore  the  angle  0  is  de- 
termined by  observing  the  change  in  the  scale  reading  when 
the  needle  is  deflected. 

The  intensity  of  the  magnetic  field  at  the  center  of  a  coil  of 
n  turns  is,  from  Chapter  XI, 

_ 


where  r  is  the  mean  radius  of  the  coil. 

Since  the  coil  of  the  tangent  galvanometer  is  so  placed  that 
H'  is  at  right  angles  to  the  earth's  field,  H,  the  angle  which  the 
resultant  of  these  two  makes  with  the  latter  is  given  by  the 
expression, 

H'      2wln 
tan  e---^.    —a, 

whence, 

/  =  ~  —  tan  6. 

2irn 

where  I  denotes  the  value  of  the  current  in  C.G.S.  units  and 
0  is  the  corresponding  deflection  of  the  needle. 

76.  The  Coulometer.  —  Another  effect  of  an  electric  current 
is  manifest  when  the  current  passes  through  an  electrolyte, 
such  as  a  solution  of  copper  sulphate  in  water.  At  the  point 
where  the  current  leaves  the  solution  copper  is  deposited,  and 
the  amount  of  copper  so  deposited  is  directly  proportional  to 
the  current  and  the  time  which  it  flows,  that  is,  to  the  number 
of  coulombs  of  electricity  that  have  passed  through  the 
electrolyte.  With  proper  precautions  this  gives  a  good  method 
for  measuring  current,  and  indeed  the  value  of  the  International 
Ampere  is  fixed  in  terms  of  the  amount  of  silver  deposited  per 
second  in  a  silver  nitrate  solution. 

When  the  copper  coulometer  is  used  the  electrodes  should  be 
of  copper,  the  best  form  being  obtained  by  winding  a  meter  of 
large  copper  wire  into  a  loose  spiral  that  will  just  fit  inside 
the  glass  jar  containing  the  solution.  Another  spiral  wound 


96  ELECTRICAL  MEASUREMENTS 

smaller  and  suspended  within  the  first  one  serves  as  the  cathode. 
All  of  the  copper  surface  should  be  clean  and  never  touched 
with  the  fingers. .  When  ready  to  be  used  the  current  should  be 
passed  through  the  coulometer  for  a  sufficient  time  to  deposit  a 
uniform  layer  of  new- copper  upon  the  smaller  spiral,  which  is 
always  used  for  the  cathode.  During  this  preliminary  run 
the  current  should  be  adjusted  to  the  value  it  is  desired  to 
measure. 

As  soon  as  the  current  is  stopped  the  spiral  cathode  is 
removed  from  the  solution  and  thoroughly  rinsed  in  distilled 
water,  then  dipped  in  alcohol  to  remove  the  water.  The  alcohol 
evaporates  readily  when  the  coil  is  gently  swung  in  warm  air. 
When  completely  dry  the  cathode  is  carefully  weighed. 

When  ready  to  begin  the  run  the  cathode  is  placed  in  position 
in  the  solution,  and  at  a  given  moment  the  circuit  is  closed,  thus 
starting  the  current  at  a  known  time.  The  current  should  be 
maintained  as  constant  as  possible  for  an  hour  and  then  stopped 
at  another  known  time.  The  cathode  is  removed,  washed, 
dried  and  weighed  as  before.  The  gain  in  weight  gives  the 
amount  of  copper  deposited,  from  which  the  average  value  of 
the  current  can  be  computed.  Since  one  coulomb  will  deposit 
0.0003283  gram  of  copper,  the  current  is  given  by  the  expression, 

M 


0.0003283* 

where  M  denotes  the  mass  of  copper  deposited  in  t  seconds. 
77.  The  Kelvin  Balance. — The  Kelvin  balance  is  an  accurate 
semi-portable  instrument  for  the  measurement  of  current. 
There  are  six  flat  coils  placed  horizontally  and  through  which 
the  current  passes  in  series.  Two  of  these,  are  carried  on  a 
balanced  beam,  one  at  either  end,  while  above  and  below  each 
of  these  movable  coils  is  one  of  the  fixed  coils.  The  diagram 
shows  the  relative  positions  of  the  coils.  The  movable  coil 
cd  is  shown  in  a  vertical  section  through  a  diameter,  while  the 
dotted  lines  indicate  the  magnetic  field  in  the  same  plane  due  to 
the  fixed  coils  ab  and  ef.  It  will  be  seen  that  at  c  the  field  is 


MEASUREMENT  OF  CURRENT  97 

horizontal  and  directed  to  the  right.  If  the  current  in  the  coil 
cd  is  flowing  into  the  paper  at  c,  then  this  portion  of  the  circuit 
will  be  urged  downward.  At  the  other  side  of  the  coil  the 
direction  of  the  field  is  to  the  left,  and  the  current  at  d  is  flow- 
ing out  of  the  paper.  Hence  this  side  of  the  coil  will  also  be 
urged  downward.  The  same  is  true  for  all  parts  of  the  coil  cd, 
and  therefore  the  coil  as  a  whole  is  urged  downward  with  a 
force  proportional  to  the  product  of  the  current  it  carries  and 
the  intensity  of  the  magnetic  field  in  which  it  moves  (see 
Chapter  XI) .  The  latter  is  proportional  to  the  current  in  the 
fixed  coils,  therefore  the  downward  pull  is  proportional  to  the 
square  of  the  current  through  the  coils.  The  action  at  the 
other  end  of  the  balance  is  the  same,  but  the  direction  of  the 


. 


FIG.  42. — Coils    of   the    Kelvin   balance 

current  through  the  coils  is  such  that  the  movable  coil  ij  is 
urged  upward.     Thus  this  effect  is  added  to  the  other. 

To  restore  the  balance  a  sliding  weight  is  drawn  along  a 
graduated  beam  until  the  movable  coils  again  stand  in  their 
original  position  midway  between  the  fixed  coils.  This  posi- 
tion is  indicated  by  a  short  scale  at  the  end  over  which  moves  a 
pointer.  The  position  of  the  weight  on  the  beam  is  read  from  a 
scale  of  equal  divisions,  and  as  shown  above  this  is  proportional 
to  the  square  of  the  current.  To  obtain  the  value  of  the 
current  in  amperes  the  square  root  of  this  reading  is  multiplied 
by  the  constant  corresponding  to  the  particular  weight  used. 
There  are  four  such  weights  and  the  constants  are  0.5,  1,  2, 

7 


98  ELECTRICAL  MEASUREMENTS 

and  4,  respectively,  for  most  balances.  In  the  centi-ampere 
balance  the  result  will  then  be  given  in  centi-amperes. 

One  other  matter  must  be  noticed.  When  the  sliding  weight 
is  placed  on  the  beam  at  the  zero  end  of  the  scale  it  is  necessary 
to  place  an  equal  counterpoise  in  the  pan  at  the  other  end.  If 
this  does  not  establish  a  complete  balance  there  is  a  small  brass 
flag  carried  by  the  moving  system  which  can  be  turned  so  as  to 
throw  more  weight  to  one  side  or  the  other  as  may  be  required 
to  restore  the  balance. 

78.  The  Electrodynamometer. — The  e-lec"tro-dy"na-mom'- 
e-ter  is  an  instrument  for  measuring  currents.  It  consists 
essentially  of  two  vertical  coils,  one  fixed  in  place,  and  the 
other  free  to  turn  about  the  vertical  axis  common  to  both 
coils.  Sometimes  the  movable  coil  is  outside  the  other,  as 
in  the  Siemens  type:  in  other  forms  the  movable  coil  is  within 
the  fixed"  coil.  In  either  case  when  a  current  flows  through  the 
movable  coil  it  tends  to  turn  in  the  same  manner  as  the  coil 
of  a  D'Arsonval  galvanometer.  But  in  the  electrodynamom- 
eter  the  magnetic  field  is  not  due  to  a  permanent  steel  magnet, 
but  is  produced  by  the  current  flowing  in  the  fixed  coil.  Thus 
the  deflection  depends  upon  the  current  /  in  the  fixed  coil  as 
well  as  upon  the  current  i  in  the  movable  coil :  and  the  result- 
ing deflection  is  given  by 

il  =  A2D 

where  A2  is  a  constant  including  all  the  factors  relating  to  the 
size  and  form  of  the  coils,  etc.,  and  also  including  the  restor- 
ing couple  of  the  suspension. 

If  the  same  current  flows  through  both  coils  in  series, 

P  =  A*D        and         I  =  A  VI) 

In  the  Siemens  electrodynamometer  the  coil  is  brought  back 
to  its  resting  point  by  the  torsion  of  a  helical  spring.  D  is  the 
number  of  divisions  of  the  scale  which  measures  the  amount 
of  this  torsion. 

When  a  coil  carrying  a  current  is  suspended  in  a  magnetic 


MEASUREMENT  OF  CURRENT  99 

field,  e.g.,  the  earth's  field,  it  tends  to  turn  so  as  to  add  its 
magnetic  field  to  the  other.  If  the  electrodynamometer  is 
set  in  such  a  position  that  the  earth's  field  is  added  to  its  own, 
evidently  the  deflection  will  be  increased  by  a  corresponding 
amount.  If  the  two  fields  were  opposed  to  each  other  the 
deflection  would  be  lessened.  This  effect  can  be  eliminated 
by  turning  the  instrument  so  that  the  plane  of  the  movable 
coil  is  east  and  west. 

79.  Calibration  of  an  Electrodynamometer. —In  order  to 
use  an  electrodynamometer  for  the  measurement  of  a  current 
it  is  necessary  to  know  the  value  of  the  constant,  A,  or,  what 
is  better,  to  have  a  calibration  curve.     Such  a  curve  is  obtained 
by  joining  the  instrument  in  series  with  a  good  ammeter  or  a 
Kelvin  Balance,  and  observing  corresponding  readings  on  the 
two  instruments  when  carrying  the  same  current.     The  curve 
is  carefully  plotted,  using  currents  as  ordinates  and  the  corre- 
sponding  deflections  for  abscissae.     This  gives  a  horizontal 
parabola  passing  through  the  origin,  and  from  this  curve  the 
value  of  the  current  corresponding  to  any  deflection  can  be 
read.     With  such  a  curve,  the  dynamometer  becomes  a  direct 
reading  ammeter.     The  deflection  is  independent  of  the  direc- 
tion of  the  current  through  the  instrument,  and  therefore  it 
can  be  used  with  alternating  currents  as  well  as  with  direct 
currents.     Reversing  the  direction  of  the  current  through  one 
coil  only,  however,  will  reverse  the  direction  of  the  deflection. 

80.  For  the  measurement  of  current  by  means  of  a  standard 
cell  see  Chapter  VII. 


CHAPTER  VII 
POTENTIOMETER  METHODS 

81.  Potential  Differences. — Any  hill  may  be  called  up,  or 
down,  according  to  the  direction  in  which  one  is  going.     If 
after  traversing  several  paths  one  returns  to  the  starting-point, 
there  will  have  been  as  much  down  hill  as  up  hill  in  the  entire 
journey. 

If  going  up  hill  is  called  positive,  and  down  hill  is  called 
negative,  the  total  height,  up  and  down,  that  one  has  been 
raised  is  zero.  The  same  thing  would  be  true  if  down  hill  had 
been  called  positive  and  up  hill  called  negative,  but  it  seems 
more  natural  to  use  the  words  in  the  former  sense. 

In  this  sense,  then,  let  us  notice  that  when  we  follow  a 
stream  down  hill,  that  is,  in  the  direction  in  which  the  water 
flows,  our  change  in  level  is  negative,  and  our  final  position  is 
lower  than  that  at  which  we  started. 

82.  Kirchhoff  s  Two  Laws. — Kirchhoff  has  enunciated  in  the 
form  of  two  "Laws"  the  principle  by  which  it  is  possible  to 
investigate  the  distribution  of  current  and  potential  in  very 
complicated  cases;  cases  where  there  are  any  number  of  cells 
connected  by  a  network  of  conductors  in  any  way  whatever. 

Law  I. — If  any  number  of  conductors  meet  at  a  point,  and 
if  all  currents  flowing  to  the  point  be  considered  positive,, 
and  all  currents  flowing  from  the  point  be  considered  negative, 
then  when  the  currents  have  reached  their  steady  values  the 
algebraic  sum  of  all  the  currents  meeting  at  the  point  must 
be  zero,  or, 

I  +  /'  +  I"  +  ,  .    .    .   .    =  0. 

because  it  is  not  possible  for  a  charge  to  accumulate  indefin- 
itely at  any  point. 

100 


POTENTIOMETER  METHODS  101 

Law  II. — Let  us  suppose  that  there  is  such  a  network  of 
conductors  as  imagined  above,  with  cells  of  various  E.M.E's., 
in  the  different  branches  of  this  network;.  \  >  V  , 


If  we  imagine  ourselves  to  start  from, any  ppiuj>  jn  ,tbjg  ne,i- 
work  and  to  make  a  circuit  through  the  coriduclbiG'fcaeE  to  our 
starting-point,  we  shall  have  passed  through  conductors  of 
various  resistances,  shall  have  passed  through  various  cells 
whose  E.M.F's.  are  directed  either  in  the  direction  we  are 
going,  or  the  opposite,  and  shall  have  found  various  currents, 
some  with  us  and  some  against  us. 

As  long  as  we  are  passing  along  a  single  conductor  r,  to  which 
there  are  no  outlets,  the  current  has  some  fixed  value  i'}  but 
on  passing  a  point  where  two  or  more  conductors  meet  we 
may  find  a  different  current  if,  which  will  remain  constant 
over  the  next  piece  of  resistance  r'  up  to  the  next  place  where 
two  or  more  branches  meet.  We  can  thus  divide  our  circuit 
into  port-ions  of  resistances  r,  /,  r",  etc.,  along  each  of  which 
will  be  a  current  i,  ir,  i",  etc.,  respectively. 

If  we  call  each  product,  ri,  negative  when  we  are  following 
the  circuit  down  the  fall  of  potential,  i.e.,  with  the  current, 
and  positive  when  we  are  going  against  the  current,  and  if 
we  call  each  E.M.F.  that  we  encounter  positive  when  we  pass 
up  the  potential  difference,  i.e.,  in  the  direction  in  which  it 
tends  to  send  a.  current,  and  negative  when  we  pass  down  the 
potential  difference,  then  in  the  complete  circuit  we  must  have 

ri  +  r'i'  +  r"i"  +  etc.  +'e  +  e'  +  e"  +  etc.  =  0 

where  e,  v',  etc.,  are  the  various  E.M.F's.  that  we  pass,  and 
which  may  or  may  not  occur  in  the  resistances  r,  r',  r",  etc., 
respectively. 

Briefly,  then,  this  law  states  that  the  sum  of  all  the  potential 
differences  in  a  closed  circuit  is  zero.  This  must  be  true  since 
in  tracing  out  a  complete  circuit  we  return  to  the  starting 
point,  and  therefore  to  the  same  potential  at  which  we  started. 


102 


ELECTRICAL  MEASUREMENTS 


83.  Illustrations  of  Kirchhoff's  Second  Law.  For  a  Simple 
Circuit. — Consider  a  circuit  consisting  of  a  cell  E,  joined  in 
series  with  a  resistance  R.  Starting  at  A  and  going  around  the 
circuit  counter  clockwise  we  have  a  fall  of  potential  RI  in 
the  part  AD.  In  passing  'from  B  on  to  the  starting-point  A, 
there  is  the  further  resistance  r  of  the  battery,  and  therefore  a 
further  fall  of  potential  of  rl.  The  only  E.M.F.  in  this  circuit 
is  that  of  the  battery  and  it  tends  to  send  a  current  the  way 
we  are  tracing  out  the  circuit. 

For  this  case,  then,  the  law  gives  the  relation 

-  RI  -  rl  +  E  =  0     or    E  =  RI  +  rl. 
For  the  Wheatstone  Bridge.  — When  the  bridge  is  balanced 


1 — vwwwv 

FIG.  43. 


FIG.  44, 


• 

no  current  flows  through  the  galvanometer.     Then  applying 
Kirchhoff's  second  law  to  the  circuit  ABC  A  gives, 

PI'  _  QI"  =  0, 

since  there  is  no  E.M.F.  in  this  circuit  and  no  fall  of  potential 
in  the  galvanometer  branch  BC. 
Similarly  for  the  circuit  BDCB, 

RI'  -  SI"  =  0 

Eliminating  the  /'  and  I"  gives  the  relation 

PS  =  QR. 


POTENTIOMETER  METHODS 


103 


84.  Proof  of  Kirchhoff's  Second  Law. — In  order  to  fix  ideas 
let  us  consider  the  case  shown  in  the  figure,  and  which  may 
be  extended  to  as  many  branches  as  we  please.  The  figure 
shows  a  single  net,  or  circuit,  from  a  network  of  conductors 
through  which  currents  are  flowing.  We  will  assume  that  the 
currents  flow  as  indicated  by  the  arrows;  should  the  value  of 
any  current  be  found  negative  in  the  final  solution  it  will  mean 
simply  that  the  actual  current  flows  in  the  opposite  direction. 

Starting  at  any  point  we  please,  say  D,  and  tracing  out  the 
circuit  in  either  direction,  say  D.E.A.,  etc.,  we  can  express 


FIG.  45. 

the  differences  of  potential  over  each  branch  in  terms  of  the 
resistances  and  E.M.F's.  Since  the  current  is  taken  as 
flowing  from  E  to  D,  E  must  have  the  higher  potential,  and 
therefore  the  rise  of  potential  from  D  to  E  is  properly  expressed 
by  +  r4*4. 

Therefore  along  this  branch  of  the  circuit  the  change  in 
potential  is, 

V,  -  V,  =  +  r4*4. 

For  the  next  portion  of  the  circuit,  EA, 


104  ELECTRICAL  MEASUREMENTS 

where  in  addition  to  the  rise  of  potential,  r6i5,  there  is  the 
further  rise  of  potential  E6  as  we  pass  through  the  cell  from 
the  negative  to  the  positive  electrode.  The  internal  resist- 
ance of  the  cell  is  included  in  r5,  and  E$  denotes  the  total 
E.M.F.  of  the  cell  as  measured  by  the  condenser  method  or  by 
the  potentiometer. 
In  the  next  branch 


where  the  negative  sign  indicates  that  there  is  a,  fall  of  potential 
along  the  conductor  AB  in  the  direction  in  which  we  are  tracing 
out  the  circuit. 

For  the  branch  BC, 

73  -  72  =  -  r2t'j  -  Ez, 

because  there  is  a  drop  of  potential  as  we  pass  through  the  cell 
in  addition  to  the  fall  along  the  conductor. 
In  the  last  branch 

V,  -  73  =  -  rtit  +  #3 

since  the  cell  in  this  branch  is  set  so  that  we  pass  through  it 
from  the  negative  to  the  positive  electrode.  It  is  to  be  observed 
that  the  sign  to  be  given  E  has  nothing  to  do  with  the  direc- 
tion in  which  the  current  is  supposed  to  be  passing  through 
the  cell.  In  such  a  network  of  conductors  and  cells  it  may 
often  happen  that  the  current  will  flow  through  some  of  the 
cells  in  the  direction  opposite  to  that  which  it  would  flow  if 
such  cells  were  acting  alone. 

Adding  the  equations  for  the  complete  circuit  gives 
(V6  -  74)  +  (F!  -  76)  +  (72  -  70  +  (7,  -  72)  +  (74  -  73) 
=  r^'4  +  r6i6  +  E5  —  rtii  —  rziz  —  E2  —  r3is  +  Es  =  0 
since  the  7's  mutually  cancel  themselves. 

This  equation  is  seen  to  agree  with  the  statement  of  the 
law  as  given  in  Article  82  above. 

85.  The  Potentiometer  Method.  —  The  most  exact  method 
for  measuring  the  E.M.F.  of  a   cell   is  that   known  as  the 


POTENTIOMETER  METHODS  105 

potentiometer  method.  The  principle  employed  will  be 
readily  understood  by  referring  to  Fig.  46.  AD  represents  a 
meter  wire  and  scale  similar  to  a  slide  wire  bridge  but  of  much 
higher  resistance.  The  ends  of  this  wire  are  connected  to  the 
battery  B,  the  E.M.F.  of  which  is  greater  than  that  of  any  cell 
to  be  measured.  C  is  the  sliding  contact,  and  between  A  and 
C  can  be  included  any  fraction  of  the  fall  of  potential  along  the 
wire  from  zero  to  the  full  amount.  A  and  C  are  joined  to- 
gether through  a  shunt  circuit  containing  the  galvanometer,  a 
key  and  the  cell  whose  E.M.F.,  E,  is  to  be  measured. 

By  moving  C  along  AD  &  point  can  be  found  for  which  there 
is  zero  deflection  of  the  galvano- 
meter when  K  is  closed.    Let  a' 
be  the   distance  from  A   to    this 
point;  the  fall  of  potential   along 
this  portion  of   the  wire   is   a'pi, 
where  p  is  the  resistance  of  unit 
length  of  the  wire  and  i  is  the  cur-      FIG.  46.— Slide  wire  poten- 
rent  flowing  through  it   from   the 

battery  at  B.  Writing  Kirchhoff's  law  for  the  galvano- 
meter circuit  gives, 

W 
Er-  a'pi  =  0          or  a'pi  =  E' 

since  there  is  no  current  and  therefore  no  fall  of  potential  in 
the  galvanometer. 

Now  let  another  cell,  whose  E.M.F.,  E" ',  we  may  suppose  is 
larger  than  E',  be  substituted  for  E.  In  order  to  obtain  a 
balance  and  zero  deflection  it  will  be  necessary  to  move  C 
nearer  to  D.  Let  a"  be  the  reading  of  this  position.  Then  by 
Kirchhoff's  law,  as  before, 

a"pi  =  E" 
Dividing  one  equation  by  the  other, 

E'       a'  ,  E" 

=  -  or  E  =  a       - 


106 


ELECTRICAL  MEASUREMENTS 


If  E"  is  known,  Ef  can  be  computed  from  this  proportion, 
being  simply  a  constant  (E" /a")  times  the  scale  reading,  a'. 
By  making  the  wire  two  meters  in  length  and  adjusting  the 
current  to  give  a  fall  of  potential  of  just  two  volts  between 
A  and  D,  the  millimeter  scale  becomes  direct  reading,  1 
mm.  corresponding  to  one  millivolt.  With  this  arrangement 
the  E.M.F.  of  any  cell  can  be  read  from  the  scale  as  soon 
as  the  galvanometer  balance  is  obtained. 

86.  The  Resistance  Box  Potentiometer. — In  the  resistance 
box  potentiometer  the  wire  AD  of  the  preceding  section  is 
replaced  by  two  similar  and  well  ad- 
justed resistance  boxes;  and  instead 
of  actually  moving  C  along  the  wire, 
resistance  is  transferred  from  AC  to 
CD  by  changing  the  plugs  while  keep- 
ing the  total  resistance  unchanged. 

The  arrangement  is  shown  in  Fig. 
47.  The  cell  to  be  measured  is  placed 
at  E  where  it  is  in  series  with  a  sensi- 
tive galvanometer,  a  high  resistance, 
HR,  a  key,  and  a  resistance  box  R. 
Through  the  latter  can  be  passed  a 
small  and  constant  current  from  the 
battery  B.  In  Article  22  this  R  was 
the  voltmeter,  and  the  current  through  it  was  varied  until  the 
fall  of  potential,  Ri,  just  balanced  the  E.M.F.  of  the  cell.  In 
the  potentiometer  method  the  current  is  kept  constant  and  R 
is  varied.  Furthermore  the  galvanometer  is  a  more  sensitive 
indicator  of  when  the  balance  has  been  secured. 

The  rest  of  the  setup  consists  of  a  constant  E.M.F.  battery 
B,  for  supplying  the  constant  current,  and  a  resistance  box 
Pj  which  should  be  identical  with  R  for  convenience.  The 
total  resistance  used  in  both  P  and  R  should  be  kept  equal  to 
the  total  capacity  of  one  box;  and  if  the  boxes  are  alike  it  is 
very  easy  to  keep  the  sum  at  this  value. 

Set  up  and  test  the  arrangement,  using  first  an  old  cell  for  E 


FIG.  47. — Resistance  box 
potentiometer. 


POTENTIOMETER  METHODS  107 

so  as  not  to  endanger  a  valuable  standard  by  an  accident  or 
wrong  connection.  Keeping  R  +  P  at  the  total  amount  in  one 
box  a  balance  should  be  obtained  without  much  difficulty. 
When  it  is  certain  that  everything  is  working  correctly  the 
cells  to  be  measured  may  be  substituted  for  the  cell  at  E. 

The  fall  of  potential  across  the  resistance  R  is,  by  Ohm's 
law, 

V  =  Ri,  (1) 

where  i  is  the  current  from  the  auxiliary  battery  B.  For  zero 
deflection  of  the  galvanometer  this  fall  of  potential  must  be 
adjusted  to  just  equal  and  balance  the  E.M.F.,  E  of  the  cell 
being  measured.  That  is,  zero  deflection  means 

E  =  V,  (2) 

or,  from  (1) 

E  =  Ri.  (3) 

When  the  known  E.M.F.,  Ef,  of  another  cell  has  been 
balanced  by  the  proper  adjustment  of  Rt  this  relation  may  be 
expressed  in  the  same  way  as  before, 

E'  =  R'i  (4) 

Since  both  E'  and  R'  are  known  this  equation  determines  the 
value  of  the  current  i  as 

E' 

i  =  jp  (5) 

and  if  the  current  has  been  kept  constant  while  R  was  varied 
(by  also  varying  P  so  as  to  keep  P  +  R  constant)  and  the  bat- 
tery B  has  not  changed,  then  this  value  of  i  can  be  used 
for  the  value  of  the  current  in  (3),  giving, 

E  =  E'jp  (6) 

which  expresses  the  E.M.F.  of  the  cell  to  be  measured  in  terms 
of  known  quantities. 

A   further   simplification   can   be   introduced   as   follows: 


108  ELECTRICAL  MEASUREMENTS 

Set  R  =  10,000  E,  where  E  is  numerically  equal  to  the  E.M.F. 
of  the  standard  cell  at  its  present  temperature.  Set  P  at  its 
complementary  value  in  order  that  later  it  will  be  convenient 
to  keep  P  +  R  constant.  This,  in  general,  will  not  give  a 
balance,  but  another  resistance,  Q,  may  be  introduced  into 
the  main  circuit  and  adjusted  to  give  the  balance.  Leaving 
Q  at  this  value,  the  cell  at  E  is  replaced  by  one  whose  E.M.F. 
is  desired  and  the  balance  obtained  by  adjusting  R  and  P, 
keeping  their  sum  constant  as  before.  The  resistance  in  R 
for  a  balance  gives  directly  the  E.M.F.  of  this  cell.  In  other 
words,  the  factor  E' /R'  in  (6)  has  now  been  made  a  round 
number  ( =  0.0001000)  and  the  computation  consists  merely 
in  moving  the  decimal  point  four  places  to  the  left. 
The  data  can  be  arranged  as  below 

USE  OP  POTENTIOMETER 


Name  of  cell 

R 

P 

Q 

E 

87.  The  Potentiometer. — For  purposes  of  instruction  the 
three  box  potentiometer  just  described  enables  one  to  see 
clearly  all  parts  of  the  setup,  and  as  has  been  shown  it  is  con- 
venient to  use.  But  if  one  has  many  measurements  of  E.M.F. 
to  make,  the  mere  changing  of  the  resistances  to  obtain  the 
balances  becomes  a  laborious  task.  It  is  then  desirable  to 
have  a  quicker  and  simpler  way  of  varying  the  resistances. 
This  is  accomplished  in  the  first  place  by  arranging  the  re- 
sistances on  a  dial  over  which  turns  an  arm,  instead  of  using  a 
movable  plug;  and  in  the  second  place  there  is  some  auto- 
matic arrangement  for  keeping  constant  the  total  resistance  in 
the  circuit. 

In  a  high  resistance  potentiometer  the  resistance  R}  Fig.  47, 
is  arranged  on  the  dial  plan.  As  the  arm  sweeps  over  the  dial, 


POTENTIOMETER  METHODS  109 

thus  varying  the  amount  of  resistance  in  R,  another  arm  sweeps 
over  the  corresponding  dial  for  P,  but  in  the  opposite  direction; 
thus  by  a  single  movement  of  the  hand  the  resistances  in  R  and 
P  may  be  varied  while  their  sum  remains  constant. 

In  a  low  resistance  potentiometer  the  arrangement  is  more  as 
shown  in  Fig.  46,  where  the  resistance  of  the  main  circuit, 
BACDB,  remains  unchanged.  The  portion  A  CD  consists  of 
about  fifteen  equal  resistances,  the  first  one  being  a  long  slide 
wire.  In  making  a  balance  the  contact  C  is  moved  along  by  a 
series  of  equal  steps  until  the  balance  has  been  obtained  to  the 
nearest  step.  Then  the  final  adjustment  is  made  by  moving 
the  other  galvanometer  terminal  from  A  along  the  slide  wire 
until  the  balance  is  obtained.  In  this  arrangement  there  are 
no  variable  contacts  in  the  potentiometer  circuit,  the  contacts 
at  A  and  C  both  being  in  the  galvanometer  circuit  where  more 
or  less  resistance  does  not  affect  the  position  of  balance. 

An  arrangement  in  which  the  various  resistances  are  thus 
conveniently  brought  together  in  a  single  box  is  called  a 
potentiometer. 

88.  Standard  Cells. — In  all  measurements  of  E.M.F.  with 
the  potentiometer  it  is  necessary  to  have  a  known  E.M.F. 
which  can  be  used  as  explained  in  Article  86  (E',  Eq.  6), 
to  standardize  the  value  of  the  current  through  the  potenti- 
ometer. Such  a  known  E.M.F.  is  furnished  by  a  standard 
cell,  which  is  a  primary  battery  set  up  in  accordance  with 
definite  specifications  so  that  it  will  possess  a  definite  E.M.F. 
Such  a  cell  is  used  as  a  standard  of  E.M.F.,  and  is  never  ex- 
pected to  furnish  a  current.  Since  the  E.M.F.  of  a  cell  de- 
pends upon  the  materials  used  in  its  construction,  and  not 
at  all  upon  its  size,  standard  cells  are  made  small,  both  for 
economy  of  materials  and  for  convenience  in  handling  them. 
A  simple  form  of  a  standard  cell  consists  of  a  small  Daniell 
cell  of  the  porous  cup  form,  set  up  with  half  saturated  solu- 
tions of  zinc  sulphate  and  of  copper  sulphate.  The  copper 
electrode  should  be  clean  and  the  zinc  electrode  well  amalga- 
mated. This  cell  has  an  E.M.F.  of  about  1.08  volts. 


110  ELECTRICAL  MEASUREMENTS 

89.  The  Western  Standard  Cell. — The.Weston  standard  cell 
was  devised  by  Mr.  Edward  Weston.     It  has  been  the  sub- 
ject of  much  study  and  investigation,  so  that  now  it  is  possible 
for  investigators  in  different  parts  of  the  world  to  set  up  such 
cells  and  know  that  they  will  have  the  same  E.M.F.  to  within 
less  than  a  ten-thousandth  of  a  volt.     In  order  to  attain  such 
accuracy  it  is  necessary  that  the  cells  be  set  up  in  strict  ac- 
cordance   with    the    specifications,    using    only    the    purest 
materials. 

The  cell  is  usually  set  up  in  an  H-shaped  glass  vessel  having 
dimensions  of  a  few  centimeters.  At  the  bottom  of  one  leg  is 
placed  pure  mercury  to  form  the  positive  electrode  of  the  cell. 
Connection  to  this  is  made  by  a  fine  platinum  wire  sealed  into 
the  glass,  the  inner  end  being  completely  covered  by  the  mer- 
cury. At  the  bottom  of  the  other  leg  there  is  placed,  similarly, 
some  cadmium  amalgam  which  when  warm  can  be  poured  in 
like  the  mercury  and  then  hardens  as  it  cools.  A  second  plati- 
num wire  through  the  glass  at  the  bottom  of  the  leg  makes 
electrical  connection  with  this  electrode.  The  electrolyte 
is  a  saturated  solution  cf  cadmium  sulphate,  containing 
crystals  of  cadmium  sulphate  in  order  to  keep  the  solution 
saturated  at  all  times.  The  mercury  electrode  is  protected 
from  contamination  by  the  cadmium  in  this  solution  by  a 
thick  layer  of  a  paste  consisting  mainly  of  mercurous  sulphate. 
Cadmium  ions  from  the  solution  coming  through  this  paste 
form  cadmium  sulphate,  and  only  mercury  ions  pass  on  and 
come  into  contact  with  the  mercury  electrode.  This  paste  is 
thus  an  efficient  depolarizer.  (See  page  9.) 

90.  Use  of  a  Standard  Cell. — A  cell  set  up  as  described  above 
will  have  a  definite  E.M.F.  which  varies  but  slightly  with  the 
temperature.     Its  internal  resistance  is  high,  and  therefore  it 
would  not  be  possible  for  it  to  furnish  much  current.     In  fact, 
any  appreciable  current  drawn  from  the  cell  would  polarize  it 
somewhat,  thereby  decreasing  its  E.M.F.  by  an  unknown 
amount  and  thus  destroy  the  only  value  which  the  cell  pos- 
sesses.    The  depolarizer  tends  to  restore  the  E.M.F.  to  its 


POTENTIOMETER  METHODS  111 

original  value,  but  the  time  required  depends  upon  the  amount 
of  polarization. 

Standard  cells  may  be  used  to  charge  condensers  to  a  known 
difference  of  potential,  for  in  this  case  there  is  no  steady  cur- 
rent drawn  from  the  cell  and  the  transient  current  is  not 
sufficient  to  cause  an  appreciable  polarization.  When  used  in 
connection  with  a  potentiometer  the  cell  should  always  be 
placed  in  series  with  a  sensitive  galvanometer.  If  it  is  neces- 
sary to  reduce  the  deflection  this  should  be  done  by  means 
of  a  high  resistance  in  series  with  the  cell,  instead  of  a  shunt 
on  the  galvanometer  which  would  still  allow  the  large  cur- 
rent to  flow  through  the  standard  cell.  When  using  the  gal- 
vanometer the  key  should  be  lightly  and  quickly  tapped  so  as 
to  give  a  deflection  of  only  a  centimeter  or  two.  This  will 
indicate  the  direction  of  the  current  as  clearly  as  a  larger  deflec- 
tion and  does  not  injure  the  standard  cell. 

91.  Comparison  of  Resistances  by  the  Potentiometer. — One 
of  the  accurate  methods  for  comparing  two  resistances,  par- 
ticularly when  these  are  not  very  large,  is  by  means  of  a  poten- 
tiometer. The  two  resistances  to  be  compared  are  joined  to- 
gether in  series  with  a  battery  and  sufficient  other  resistance  to 
ensure  a  steady  current.  This  current  should  not  be  large 
enough  to  change  the  resistances  by  heating  them.  Other 
things  being  equal,  it  is  desirable  to  have  the  fall  of  potential 
over  each  resistance  about  one  volt.  Let  the  two  resistances 
be  denoted  by  S  and  T;  then  the  fall  of  potential  over  each  will 
be  SI  and  T7,  respectively. 

The  actual  measurements  are  very  simple.  With  the 
potentiometer  set  up  as  used  for  the  comparison  of  E.M.F's., 
the  fall  of  potential  over  each  resistance  is  measured.  Let 
the  readings  on  the  potentiometer  be  R'  and  R",  respectively. 
Then  from  the  conditions  of  balance, 

TI  =  R'i        and        SI  =  R"i, 

where  i  denotes  the  current  through  the  potentiometer 
resistances. 


112  ELECTRICAL  MEASUREMENTS 

From  this  it  follows  at  once  that 


E 


and  this  relation  can  be  determined  as  accurately  as  the 
potentiometer  measurements  can  be  made. 

92.  Calibration  of  a  Voltmeter. — The  voltmeter  is  joined  to 
a  storage  battery  or  other  source  which  will  maintain  the  de- 
flection steady  at  the  point  it  is  desired  to  calibrate.  If  the 
voltmeter  is  direct  reading  it  should  read  the  difference  of 
potential  between  its  own  binding  posts.  To  calibrate  the 

scale,  then,  it  is  only  necessary  to 
measure  this  same  difference  of  po- 
tential by  some  precise  method  and 
compare  this  measured  value  with 
the  reading  of  the  voltmeter. 

In  parallel  with  the  voltmeter 
is  joined  a  circuit  consisting  of  two 
resistance  boxes,  A  and  B,  and  in 
parallel  with  A  is  a  circuit  contain- 
ing the  galvanometer,  a  standard 
cell,  a  high  resistance  and  a  key.  It 
is  best  to  have  about  a  thousand 
ohms  in  A  and  B  together  for 
each  volt  read  by  the  voltmeter.  A 

preliminary  calculation  will  give  the  resistance  which  should  be 
placed  in  each  A  and  B  to  give  a  fall  of  potential  over  A  about 
equal  to  the  E.M.F.  of  the  standard  cell.  When  these  ap- 
proximate resistances  have  been  placed  in  A  and  B,  the  key, 
K,  can  be  quickly  and  cautiously  tapped  and  the  direction  of 
the  deflection  noted.  In  using  a  standard  cell  in  this  way  as 
little  current  as  possible  should  be  allowed  to  flow  through  it. 
Even  a  slight  polarization  will  lower  its  E.M.F.  by  an  unknown 
amount  and  then  it  can  no  longer  be  called  a  "standard"  cell. 
Furthermore,  nothing  is  gained  by  deflecting  the  galvanometer 
"off  the  scale,"  as  a  deflection  of  a  centimeter  or  two  takes  less 


FIG.  48. — Calibration  of  the 
voltmeter,  Vm. 


POTENTIOMETER  METHODS  113 

time  and  is  fully  as  definite  as  the  larger  deflection.  The  high 
resistance  is  inserted  for  this  very  protection  of  the  cell  and 
therefore  is  better  than  a  shunt  on  the  galvanometer. 

By  adjusting  the  values  of  A  and  B  the  galvanometer  deflec- 
tion can  be  reduced  to  zero.  The  high  resistance  can  be  short 
circuited  for  the  final  balance  if  the  deflections  are  small. 
When  thus  balanced,  Kirchhoff's  law  gives  for  the  circuit 
through  the  voltmeter  and  A  and  B, 

Ai  +  Bi  -  SI  =  0, 

where  SI  is  the  fall  of  potential  over  the  voltmeter  and  is  what 
should  be  indicated  on  its  scale. 

Similarly  for  the  circuit  through  the  standard  cell 

Ai  =  E 

Eliminating  i  by  division, 


If  V  is  the  reading  of  the  voltmeter,  then 

V  +  c  =  81, 

and  the  correction  to  be  applied  at  this  point  is, 

c  =  SI  -  V 

The  computations  can  be  greatly  reduced  if  A  is  kept  at 
the  value  A  =  1000  E.  Thus  if  the  E.M.F.  of  the  standard 
cell  is  1.018  volts  at  the  temperature  of  the  room,  A  would 
then  be  set  at  1018  ohms  and  all  of  the  adjustment  made  by 
changing  B.  The  values  of  SI  are  then  given  directly  by  the 
sum  of  A  and  B  divided  by  1000. 

Other  points  on  the  scale  can  be  obtained  by  using  a  differ- 
ent number  of  cells  in  the  main  battery,  or,  if  the  voltmeter 
is  low  reading,  by  adding  some  resistance  in  series  with  the 
battery.  If  the  battery  cannot  be  divided  into  a  sufficient 
number  of  steps  a  useful  method  is  to  join  across  the  battery 
terminals  a  high  resistance  rheostat  and  connect  the  voltmeter 

8 


114 


ELECTRICAL  MEASUREMENTS 


to  one  terminal  and  to  the  sliding  contact.  Any  desired  vol- 
tage can  then  be  obtained  up  to  the  maximum  of  the  battery 
by  simply  sliding  the  movable  contact  along  the  rheostat.  It 
is  evident  that  this  method  can  not  be  used  to  calibrate  points 
below  the  E.M.F.  of  the  standard  cell. 

A  calibration  curve  should  be  plotted,  with  voltmeter  readT 
ings  as  abscissae  and  the  corresponding  corrections  as  ordi- 
nates.  If  there  is  a  "zero  correction"  because  the  needle 
does  not  indicate  zero  correctly,  such  correction  should  be  made 
before  computing  the  calibration  corrections. 

93.  A  More  Convenient  Method. — In  case  the  voltmeter 
reading  can  be  varied  continuously,  or  by  very  small  steps, 

either  by  means  of  a  sliding  rheo- 
stat as  shown  in  Fig.  48,  or  by 
means  of  a  resistance  box  in  series 
with  the  voltmeter  as  shown  in  Fig. 
49,  it  will  be  found  more  expeditious 
to  set  A  and  B  at  the  values  corres- 
ponding to  a  given  voltage:  and 
then  by  varying  the  rheostat  to 
bring  the  voltmeter  to  this  voltage, 
the  balance  being  indicated  by  the 
galvanometer  the  same  as  before. 
The  voltmeter  reading  should  give 
this  same  voltage;  if  it  does  not, 
the  discrepancy  is  the  correction 
which  must  be  applied  to  the  voltmeter  reading  at  this  point. 

94.  Calibration  of  a  Low  Reading  Voltmeter. — A  low  reading 
voltmeter  can  be  readily  calibrated  by  the  potentiometer 
method  described  in  Article  86  above.     The  voltmeter  is 
connected  to  a  battery  of  one  or  two  good  dry  cells  with  a 
resistance  box,  r,  in  series.     By  varying  the  amount  of  resist- 
ance in  r,  the  pointer  of  the  voltmeter  can  be  brought  to  any 
desired  part  of  the  scale.     The  voltmeter,  while  still  connected 
with  its  battery,  is  also  inserted  in  the  galvanometer  circuit 
of  the  potentiometer  in  place  of  the  standard  cell.    There 


r~  —  i 
p 

Q 

\ 

B 

FIG.  49. — Cabralition  of  tHe 
milli-voltmeter,  Vm. 


POTENTIOMETER  METHODS  115 

will  thus  be  two  electric  circuits,  each  with  its  own  battery  ; 
one  battery  supplying  the  current  through  r  and  the  volt- 
meter, while  the  potentiometer  current  through  P  and  Q 
comes  from  the  battery  B.  The  galvanometer  circuit  con- 
nects these  two  circuits,  and  the  current  through  it  may  be  in 
either  direction,  or  made  zero  by  adjusting  the  resistance 
in  P. 

Applying  Kirchhoff's  second  law  to  the  galvanometer 
circuit  gives 

8(1"  +  i)  +  gi  -  P'(I'  -  i)  =  0, 

where  8  is  the  resistance  of  the  voltmeter.  For  no  current  in 
the  galvanometer, 

SI"  =  PT 

The  value  of  the  constant  current  I'  is  determined  by  using 
a  cell  of  known  E.M.F.  in  the  same  manner  as  explained  in 
Article  86  (eq.  5).  The  voltmeter,  with  its  battery,  is 
removed  from  the  galvanometer  circuit  by  leaving  K'  open, 
and  the  standard  cell  is  inserted  in  the  same  circuit  by  using 
the  key  K  as  shown  in  Fig.  49.  The  resistance  in  P  is  now 
adjusted  to  a  new  value  P"  to  give  no  current  through  the 
galvanometer.  Then, 

E  =  P'T 

since  by  keeping  P-\-  Q  constant  the  current  is  the  same  as 
before.  Therefore, 

81"  -EJ? 

and  this  is  what  the  voltmeter  should  read.  If  the  voltmeter 
reading  is  V,  the  correction  to  be  applied  is 

The  correction  curve  should  be  drawn  with  the  observed 
voltmeter  readings  for  abscissae  and  the  corresponding 
corrections  for  ordinates. 

95.  Measurement  of  Current  by  Means  of  a  Standard  Cell. 
— A  current  can  be  measured  quickly  and  approximately  by 


116 


ELECTRICAL  MEASUREMENTS 


reading  an  ammeter,  through  which  the  current  is  made  to 
pass.  But  if  it  is  desired  to  measure  the  current  more  exactly 
a  more  refined  method  is  necessary;  and  the  method  always 
used  in  the  laboratory  is  to  pass  the  current,  7,  through  a  coil 
whose  resistance,  R,  is  accurately  known,  and  measure  the  re- 
sulting fall  of  potential,  RI.  This  can  be  done  by  any  method 
for  measuring  potential  differences  that  will  give  the  required 
degree  of  accuracy, 

In  the  methods  here  described  the  measurements  are  ex- 
pressed in  terms  of  a  standard  cell  either  by  the  regular  po- 
tentiometer method  described  in 
Article  86,  or  by  direct  comparison 
like  the  method  for  calibrating  a 
voltmeter. 

96.  Calibration  of  an  Ammeter. — 
For  the  calibration  of  an  ammeter 
the  instrument  is  joined  in  series 
with  a  standard  resistance.  The 
same  current  must  therefore  pass 
through  them  both.  Its  amount 
is  determined  by  measuring  the  fall 
of  potential  over  the  standard  resistance  and  this  value  is 
compared  with  the  reading  of  the  ammeter.  The  difference 
is  the  ammeter  correction.  In  the  figure  the  current  from 
the  battery  flows  in  series  through  the  ammeter,  the  standard 
resistance  R,  and  an  adjustable  resistance,  r.  In  parallel 
with  R  is  a  circuit  of  much  higher  resistance  and  consisting 
of  two  well  adjusted  resistance  boxes.  In  parallel  with  one 
of  these  is  the  galvanometer  with  a  standard  cell  and  a  high 
resistance  of  about  10,000  ohms  to  prevent  too  great  a  current 
through  the  standard  cell. 

Let  the  currents  through  the  ammeter,  standard  resistance, 
B,  and  the  galvanometer  be  denoted  by  /,  /',  i  and  i'  re- 
spectively. Applying  Kirchhoffs  law  to  the  circuit  containing 
A  and  the  galvanometer,  gives,  for  the  case  of  balance, 

E3  =  Ai. 


FIG.  50. — Calibration  of  the 
ammeter,  Am. 


POTENTIOMETER  METHODS 

At  the  same  instant  the  circuit  ABR  gives 

RF  =  (A  +  B)  i 
Solving  for  the  current  in  R 


i,  • 

''  R 


117 


Since  I  =  /'  +  i,  we  have  finally, 

r  _^l     A+B  +  R 
"  R  A 

It  is  evident  that  if  A  is  kept  at  some  round  number,  say 
5000,  and  the  adjustment  made  by  varying  B,  the  computa- 
tions will  be  much  simplified. 

If  the  reading  of  the  ammeter  is  IA  then 

/A    +    C      =    1, 

and  the  correction  to  be  added  to  this  reading  is 

c  =  /  -  7A 

CALIBRATION  OP  AMMETER  No  ..... 
Am.  Zero  =  E.  =  at  °C.  R     = 


Ammeter 

Standard  cell 

Resistances 

Correc- 

tion 

Reading 

Corrected 

Temp. 

E.M.F. 

A 

B 

I 

97.  Calibration  of  a  Mil-ammeter.  Potentiometer  Method. 
— In  this  method  small  currents  can  be  measured  as  well  as 
large  ones,  and  if  a  potentiometer  can  be  obtained,  it  is  a  more 
convenient  method  than  the  preceding.  For  the  best  results 
the  known  low  resistance,  C,  should  be  a  standard  manganin 
resistance  provided  with  permanent  potential  terminals. 


118 


ELECTRICAL  MEASUREMENTS 


The  ammeter  is  connected  in  series  with  a  storage  battery, 
a  variable  rheostat,  and  the  standard  resistance.  The  fall 
of  potential  over  the  latter  is  measured  by  the  potentiometer, 
PQ,  in  the  usual  way.  The  best  results  are  obtained  when 
this  is  about  one  volt  By  means  of  the  double  throw  switch, 
S,  either  C  or  the  standard  cell,  E,  can  be  thrown  into  the 
galvanometer  circuit.  In  the  latter  case, 
and  after  a  balance  has  been  obtained  by 
adjusting  P  until  there  is  no  deflection 
__^O — .  c  r] '  °f  ^ne  galvanometer, 

I       CG]  E  =  P'i. 


When  the  coil  C,  carrying  a  current  /, 
has  been  substituted  for  the  standard 
cell  and  a  balance  obtained  by  readjust- 
ing P  to  some  new  value,  P",  we  have, 


\ 

-I 

I 

_5i 

F 

-4                 *- 
P 

-•        » 
Q 

,1,1* 

CI  =  P"i. 


From  which, 


7_  _     * 
~~    yv          T-»f 


P* 
P' 


FIG.    51. — Calibration 
of  the  ammeter,  Am. 

This  value  for  the  current  is  com- 
pared with  the  ammeter  reading,  7A,  and  the  corresponding 
correction  is 

c  =  /  -  7A. 

A  calibration  curve  should  be  drawn,  using  the  observed 
ammeter  readings  as  abscissae  and  the  corresponding  correc- 
tions for  ordinates. 


CHAPTER  VIII 

MEASUREMENT  OF  POWER 

_  /'£)/- 

98.  The  measurement  of  electrical  power  usually  resolves 
itself  into  the  simultaneous  measurement  of  E.M.F.  and  current. 
As  stated  in  Article  11,  the  unit  of  power  is  the  Watt,  and  is 
the  power  expended  by  a  current  of  one  ampere  under  a 
potential  difference  of  one  volt.     In  Article  21,  there  was 
given  a  simple  method  for  measuring  the  power  expended  in 
a  circuit  by  the  current  from  a  battery,  using  an  ammeter 
and  a  voltmeter.     A  single  instrument  combining  in  itself 
the  functions  of  both  an  ammeter  and  a  voltmeter  is  called  a 
wattmeter.     With  such  an  instrument  the  power  may  be 
read  directly  from  a  single  scale,  in  the  same  way  as  the  current 
is  read  from  the  scale  of  an  ammeter. 

99.  The  Use  of  an  Electrodynamometer  for  the  Measure- 
ment of  Power. — An  ammeter  and  a  voltmeter  connected  as 
shown  in  Fig.  52  (a)  for  the  measurement  of  a  resistance  will 
give  at  the  same  time  the  power  expended  in  R.    Let  B  denote 
the  source  of  the  current.     The  voltmeter,  Vm,  measures  the 
fall  of  potential,  E,  between  the  terminals,  while  the  ammeter, 
Am,  gives  the  value  of  the  current.     The  product,  El  =  W, 
gives  the  power  in  watts. 

This  result  can  be  expressed  in  a  different  form.  If  in  place 
of  a  direct  reading  voltmeter  there  had  been  a  large  resistance 
of  S  ohms  in  series  with  a  mil-ammeter  for  measuring  the 
current,  i,  through  it,  then 

E  =  Si,       and       W  =  Sil. 

In  this  form  it  is  seen  that  the  measurement  of  power 
implies  the  product  of  two  currents;  and  in  Article  78  it  was 

119 


120 


ELECTRICAL  MEASUREMENTS 


seen  that  an  electrodynamometer  is  an  instrument  for  measur- 
ing the  product  of  two  currents.  Therefore  an  electro- 
dynamometer  can  be  used  as  a  wattmeter  if  it  is  connected 
into  the  circuit  in  the  proper  manner  for  this  purpose. 

Let  R,  Fig.  52  (c),  be  the  circuit  in  which  the  power  is  to  be 
measured.  The  low  resistance  coil,  a,  of  the  wattmeter  W  is 
connected  in  series  with  R  as  was  the  ammeter  of  Fig.  52  (a). 
The  other  coil,  v,  is  joined  in  series  with  a  resistance  of  several 
hundred  ohms  to  form  a  shunt  circuit  of  high  resistance,  and 
this  is  connected  in  the  place  of  the  voltmeter  to  measure  the 
fall  of  potential  over  R  and  a.  For  let  i  denote  the  value 
of  the  current  through  this  shunt  circuit,  and  S  its  resistance. 


w 


FIG.  52. — Measurement  of  power. 

The  fall  of  potential  is  then  Si,  as  in  Fig.  52  (6).  This  current 
i  through  one  coil  of  the  instrument,  together  with  the  main 
current  I  through  the  other  coil,  will  produce  a  deflection  D, 
proportional  to  the  product  of  the  two  currents.  From  the 
equation  of  the  electrodynamometer, 

il  =  A2D, 

where  A  is  the  same  constant  that  was  previously  determined. 
Since  the  power  being  expended  in  R  is  W  =  Sil,  we  now 
have, 

W  =  Sil  =  SA2D. 


If  the  constant  of  the  instrument   is  known,   then  SA2 
becomes  the  factor  for  reducing  the  scale  readings  to  watts. 


MEASUREMENT  OF  POWER  121 

In  case  this  factor  is  unity,  as  it  can  be  made  by  adjusting  the 
value  of  S,  the  wattmeter  is  said  to  be  direct  reading. 

It  may  be  that  the  value  of  A  is  not  known,  but  instead  there 
is  a  calibration  curve  for  the  instrument  when  used  as  an 
electrodynamometer.  In  this  case  the  value  of  A2D  can  be 
obtained  directly  from  the  curve,  for  it  is  the  square  of  the 
current  /'  which  would  produce  the  same  deflection,  D. 

Thus  the  power  expended  in  R  is 

W  =  SI'2, 

where  S  is  the  resistance  of  the  shunt  circuit  and  /'  is  not  any 
real  current,  but  it  is  the  current  which  gave  the  same  de- 
flection when  the  instrument  was  used  as  an  electrodyna- 
mometer  and  the  value  of  which  can  be  obtained  from  the  cali- 
bration curve. 

100.  The  Weston  Wattmeter. — The  Weston  wattmeter 
consists,  essentially,  of  a  moving  coil  electrodynamometer. 
The  fixed  coil  is  wound  in  two  sections  on  a  long  cylindrical 
tube,  within  and  between  which  is  the  movable  coil.  The 
latter  is  wound  with  fine  wire  upon  a  very  short  section  of  a 
cylindrical  tube  of  somewhat  smaller  diameter  than  the  fixed 
coil,  and  supported  on  pivots  so  that  it  can  readily  turn  about 
its  vertical  diameter.  Attached  to  the  movable  coil  is  a  long 
light  pointer  which  moves  over  a  graduated  scale. 

In  the  position  of  rest  the  axis  of  the  movable  coil  makes  an 
angle  of  about  45°  with  the  axis  of  the  fixed  coil.  When 
deflected  so  that  the  pointer  is  at  the  middle  of  the  scale  the 
two  coils  are  at  right  angles.  At  the  extreme  end  of  the 
scale  the  coil  stands  at  45°  on  the  other  side  of  the  symmetrical 
position.  This  gives  a  fairly  uniform  scale  over  its  entire 
length.  A  spiral  spring  brings  the  coil  to  the  zero  position 
and  provides  the  torque  necessary  to  balance  the  electro- 
dynamic  couple  due  to  the  currents  in  the  two  coils. 

In  addition  to  the  main  fixed  coil  there  is  another  fixed 
coil  of  fine  wire  and  having  the  same  number  of  turns  as  the 
other,  so  that  a  current  sent  through  one  coil  and  back  through 


122 


ELECTRICAL  MEASUREMENTS 


the  other  will  produce  no  magnetic  field  at  the  place  of  the 
movable  coil.  It  is  then  possible  to  compensate  for  the  effect 
of  the  shunt  current  passing  through  the  series  coil,  for  the 
shunt  current  can  be  lead  back  through  this  second  coil  and  thus 
be  made  to  neutralize  its  action  upon  the  movable  coil.  When 
it  is  desired  not  to  use  the  compensation  coil  it  is  replaced  by  an 
equal  resistance,  this  connection  being  brought  out  to  a  third 
binding  post. 


FIG.  53. 

In  the  wattmeter  reading  up  to  150  watts  the  resistance  of 
the  series  coil  is  0.3  ohm,  and  that  of  the  shunt  circuit  is  2600 
ohms.  The  compensation  winding  is  about  3  ohms. 

101.  Comparison  of  a  Wattmeter  with  an  Ammeter  and  a 
Voltmeter. — The  reading  of  a  wattmeter  can  be  compared  with 
the  power  measured  by  an  ammeter  and  a  voltmeter,  provided 
that  the  latter  instruments  are  connected  to  measure  precisely 


MEASUREMENT  OF  POWER  123 

the  same  power  as  the  wattmeter.  This  means,  for  the  un- 
compensated  wattmeter,  that  the  current  through  the  series 
coil  of  the  wattmeter  must  be  measured  by  the  ammeter,  and 
the  voltmeter  must  be  connected  so  as  to  measure  the  same  fall 
of  potential  as  the  shunt  coil  of  the  wattmeter. 

This  is  accomplished  by  the  connections  shown  in  Fig.  53. 
The  power  thus  measured  is  not  that  expended  in  R  alone,  but 
it  includes  the  power  expended  in  .the  voltmeter  and  in  the 
shunt  circuit  of  the  wattmeter.  But  since  both  instruments 
measure  this  same  power,  the  reading  of  the  wattmeter  should 
agree  with  the  product  of  the  readings  from  the  ammeter  and 
voltmeter. 

If  the  wattmeter  is  compensated  so  that  the  power  measured 
by  it  does  not  include  the  power  expended  in  its  own  shunt  cir- 
cuit, then  the  ammeter  must  be  connected 
so  as  not  to  measure  this  shunt  current. 
But  as  it  is  not  possible  to  connect  the 
ammeter  and  voltmeter  so  as  not  to  meas- 
ure 'the  power  expended  in  one  or  the  other 
of  them,  the  best  arrangement  will  be  to  join 
the  voltmeter  in  parallel  with  R  as  shown  F  _. 

in  Fig.  54.  The  power  measured  by  the 
ammeter  and  voltmeter  will  be  that  expended  in  both  R  and 
the  voltmeter;  that  measured  by  the  wattmeter  will  be  greater 
than  this  by  the  amount  of  power  expended  in  the  ammeter. 
The  latter  can  be  computed  from  the  formula  r/2,  and  added 
to  the  product  of  the  readings  from  the  ammeter  and  volt- 
meter. With  this  slight  correction  the  wattmeter  reading 
should  equal  VI. 

102.  Power  Expended  in  a  Rheostat. — a.  When  carrying  a 
constant  current.  The  object  of  this  exercise  is  to  give  the 
student  some  personal  experience  in  the  measurement  of  power 
and  in  the  careful  use  of  a  variable  rheostat.  In  this  first  part 
a  variable  resistance  is  joined  in  series  with  a  large  E.M.F. 
and  considerable  other  resistance,  so  that  the  variations  in  r 
will  not  materially  change  the  value  of  the  current  through  the 


124  ELECTRICAL  MEASUREMENTS 

circuit.  If  desired  this  variable  resistance  may  include  an 
ammeter,  and  the  current  may  be  kept  always  at  the  same 
value. 

The  wattmeter  is  connected  to  r  as  shown  in  the  figure,  and 
readings  of  the  power  expended  in  the  rheostat  are  taken  for  the 
entire  range  of  the  resistance.  If  the  values  of  the  latter  are 
not  known  they  can  be  measured  by  one  of  the  methods  pre- 
viously given.  A  curve  should  be  drawn,  using  the  resistances 
in  the  rheostat  as  abscissae  and  the  corresponding  amounts  of 
power  as  ordinates.  The  report  should  contain  a  discussion 
explaining  why  this  curve  comes  out  with  the  form  it  has. 

b.  When  under  constant  voltage.     In  this  part  of  the  exercise 
the  arrangement  is  much  the  same  as  before,  except  that  the 
high  E.M.F.  is  replaced  by  a  few  cells  of 
a  storage  battery,  and  r  is  now  the  only 
resistance  in  the  circuit.     Starting  with  the 
largest  values  of  r,  readings  of  the  watt- 
meter  are    taken    and  plotted  as  before. 
It  will   not  be  safe  to  reduce  r  to    zero, 
FIG.  55.  and  readings  should  be  continued  only  for 

current  values  that  are  not  too  large  for 
the  apparatus  used.  This  portion  of  the  report  should  give  a 
discussion  of  what  would  probably  happen  if  the  rheostat 
were  reduced  to  zero. 

102a.  Efficiency  of  Electric  Lamps. — An  interesting  exer- 
cise, and  one  that  furnishes  considerable  information,  is  the 
determination  of  the  efficiency  of  various  types  of  electric 
lamps.  By  means  of  a  wattmeter  the  amount  of  electrical 
energy  supplied  to  the  lamp  is  readily  measured.  The  light 
that  the  lamp  furnishes  can  be  measured  by  a  photometer. 
The  efficiency  of  the  lamp  is  expressed  as  the  ratio  of  the 
candle  power  of  the  lamp  to  the  electrical  power  expended, 
and  is  given  as  so  many  "  candles  per  watt."  Thus  the  ef- 
ficiency is  not  an  abstract  ratio,  as  in  most  cases,  because  it 
is  not  possible  to  measure  light  in  watts.  But  this  does  not 
prevent  a  satisfactory  comparison  of  different  lamps. 


MEASUREMENT  OF  POWER  125 

Any  good  form  of  photometer  can  be  used  to  measure  the 
candle  power  of  the  lamp  being  examined.  If  none  is  at  hand, 
a  simple  form  can  be  made  by  standing  some  object  in  front 
of  a  white  screen.  The  standard  light  and  the  one  being 
measured  will  each  cast  a  shadow  of  the  object  on  the  screen, 
and  by  varying  the  distance  of  one  of  the  lights  from  the 
screen  the  intensities  of  the  two  shadows  can  be  made  equal. 
The  intensities  of  the  two  lights  are  then  to  each  other  as  the 
squares  of  their  respective  distances  from  the  screen. 

If  a  number  of  different  kinds  of  lamps  are  available  it  is 
instructive  to  measure  the  efficiency  of  each  one.  Tungsten 
lamps  can  be  compared  with  carbon  lamps,  and  lamps  that 
have  been  in  use  for  a  long  time  can  be  compared  with  new 
lamps  of  the  same  kind.  It  is  also  interesting  to  determine  the 
efficiency  of  the  same  lamp  when  burned  at  different  voltages, 
and  it  is  well  to  plot  a  curve  with  efficiencies  as  ordinates 
with  the  corresponding  voltages  for  abscissae.  Incidentally 
the  curve  showing  the  variation  of  candle  power  with  voltage 
can  be  plotted. 

103.  Measurement  of  Power  in  Terms  of  a  Standard  Cell. — 
In  the  preceding  chapters  there  have  been  given  methods  for 
measuring  either  a  current  or  a  difference  of  potential  in  terms 
of  the  E.M.F.  of  a  standard  cell.  By  combining  two  of  these 
methods  it  is  possible  to  measure  power  in  like  manner,  and 
this  is  especially  useful  when  it  is  desired  to  know  accurately 
the  value  of  a  given  amount  of  power.  For  example,  in  some 
methods  of  calorimetry  it  is  necessary  to  have  supplied  a  known 
amount  of  heat.  Often  the  actual  amount  is  not  essential,  but 
whatever  it  is  it  must  be  known  to  a  high  degree  of  accuracy. 
In  such  cases  it  is  convenient  to  generate  the  heat  by  an  elec- 
tric current  flowing  through  a  resistance  coil,  and  then  to  meas- 
ure the  electrical  power  expended  in  terms  of  a  standard  cell 
of  known  E.M.F.  This  means  the  measurement  of  both  the 
current  and  the  fall  of  potential,  as  the  resistance  of  the  coil 
usually  can  not  be  accurately  determined  under  the  conditions 
of  actual  use. 


126 


ELECTRICAL  MEASUREMENTS 


One  convenient  arrangement,  which  is  capable  of  wide 
variation  in  the  amount  of  power  that  can  be  measured,  is 
shown  in  the  figure.  The  heating  coil  in  which  the  power  is 
expended  is  denoted  by  H.  In  series  with  this  is  a  coil  C,  of 
sufficient  current  carrying  capacity  not  to  be  heated  by  the 
current  through  it.  There  is  also  a  variable  resistance,  r, 
by  which  the  current  can  be  brought  to  any  desired  value. 
The  current  through  C  is  measured  by  the  method  for  cali- 
brating an  ammeter  (see  Article  96) 
and  the  fall  of  potential  over  H  is 
measured  by  the  method  for  cali- 
brating a  voltmeter  (see  Article  92) . 
The  standard  cell  is  joined  in  se- 
ries with  a  sensitive  galvanometer 
and  a  high  resistance.  The  circuit 
thus  formed  is  connected  to  the 
middle  of  a  double  throw  switch  S. 
One  end  of  this  switch  is  connected 
to  B,  which  is  a  portion  of  a  shunt 
around  H.  By  adjusting  A  and  B, 
the  fall  of  potential  over  the  latter 

FIG.  56.— Measurement  of    can  be  made  equal  to  the  E.M.F.  of 
power  expended  in  H  in  terms     ,.  .      ,       „  ,  , 

of  a  standard  cell,  E.  the  standard  cell,  as  shown  by  no 

deflection  of  the  galvanometer  when 

the  key  K  is  closed.     The  total  fall  of  potential  over  H  is, 
then, 

A    _1_    R 

V   = 


B 

In  the  same  way  there  is  a  shunt  circuit,  PQ,  in  parallel 
with  C,  and  the  part  Q  is  joined  to  the  other  end  of  the  double 
throw  switch.  When  the  switch  is  thrown  to  this  side,  P 
and  Q  can  be  adjusted  to  give  no  deflection  of  the  galvanometer 
when  the  key  is  closed.  This  means  that  the  fall  of  potential 
over  Q  has  been  made  the  same  as  the  E.M.F.  of  the  standard 
cell,  and  the  total  fall  over  C  is  computed  as  shown  above 


MEASUREMENT  OF  POWER  127 

for  H.    This,  divided  by  the  resistance  C,  gives  the  current 
through  C  as 

ESP  +  Q 
~~  C       Q 

A  little  consideration  will  show  that  the  main  current 
through  the  battery  is  larger  than  I  by  the  amount  of  current 
that  flows  through  the  shunt  circuit  PQ;  and  the  current 
through  H  is  smaller  than  the  main  current  by  the  amount  of 
current  that  flows  through  the  shunt  AB.  If  Q  is  set  equal 
to  B,  then  since  the  fall  of  potential  over  each  is  the  same,  the 
currents  through  these  two  shunts  will  be  equal.  There- 
fore the  current  through  H  will  be  7,  the  current  through  C. 
The  power  expended  in  H  is,  then, 

w  _  VI  - 

~ 


B2  C 

when  the  two  shunt  currents  have  thus  been  made  equal. 
104.  Calibration  of  a  Non-compensated  Wattmeter.  —  The 

wattmeter  in  this  case  may  be  an  electrodynamometer  with 
two  separate  coils,  or  it  may  be  the  regular  Weston  wattmeter 
used  without  the  compensation  coil.  The  series  coil  of  the 
wattmeter  is  connected  in  series  with  a  resistance,  H,  in  which 
can  be  expended  the  power  measured  by  the  wattmeter.  There 
is  also  in  series  a  standard  resistance,  C,  whose  value  is  accu- 
rately known  and  which  has  sufficient  current  carrying  capa- 
city not  to  be  heated  by  the  currents  used  in  the  calibration. 
A  variable  resistance,  r,  and  a  storage  battery,  z,  complete  the 
main  circuit.  The  shunt  coil,  ab,  of  the  wattmeter  is  connected 
in  parallel  with  H.  Two  well-known  resistances,  A  and  B,  are 
also  connected  in  parallel  with  H.  The  power  measured  by 
the  wattmeter  is  then  the  total  power  expended  in  H,  the 
wattmeter  shunt  circuit,  and  the  circuit  consisting  of  A  and  B. 
This  power  is  the  product  of  the  current  through  these  three 
in  parallel  and  the  potential  difference  between  m  and  n. 
Both  of  these  quantities  are  determined  by  the  potentiometer. 


128 


ELECTRICAL  MEASUREMENTS 


The  potentiometer  is  represented  by  the  three  resistances, 
P,  Q,  and  R.  Across  P  is  joined  the  galvanometer  and  stand- 
ard cell  in  the  usual  way.  When  P  has  been  adjusted  for  a 

balance, 

E 

where  E  is  the  E.M.F.  of  the  standard  cell  and  i  is  the  steady 
constant  current  that  is  flowing  through  the  main  circuit  of 
the  potentiometer. 


FIG.  57. — Calibration  of  the  wattmeter,  W. 

To  determine  the  value  of  the  current  through  the  standard 
resistance,  wires  are  brought  from  the  terminals  of  C  to  the 
double  throw  switch  S.  When  this  switch  is  thrown  up  and  Kf 
is  closed,  C  is  connected  into  the  galvanometer  circuit  in  the 
place  of  the  standard  cell.  Readjusting  the  potentiometer 
for  a  new  balance,  P',  gives, 

EP' 


1C  =  iP' 


and  hence, 


CP' 


MEASUREMENT  OF  POWER  129 

To  determine  the  fall  of  potential  over  H  the  double  throw 
switch  is  thrown  down,  thus  connecting  the  resistance  A  into 
the  galvanometer  circuit.  Let  i'  denote  the  value  of  the  small 
shunt  current  flowing  through  A  and  B.  Then  adjusting  the 
potentiometer  for  a  balance, 

.  i'A  =  iP"  =  ^f, 
where  P"  is  the  new  reading  of  the  potentiometer.    The  fall 

A       I      D 

of  potential  over  both  A  and  B  is  — -j —  times  as  large,  or, 

T/  _  EP»  A  +  B 
P        A 

and  this  is  the  same  as  the  fall  of  potential  over  H. 
The  power  measured  by  the  wattmeter  is,  then, 

W,_VT_E*P'P'A+B 
~  P*     C        A 

If  the  reading  of  the  wattmeter  is  W,  the  correction  to  be 
added  to  this  reading  is, 

c  =  W'  -  W. 

Different  readings  of  the  wattmeter  are  secured  by  chang- 
ing the  current  through  H.  A  calibration  curve  can  be 
plotted,  using  the  readings  of  the  wattmeter  as  abscissae  and 
the  corresponding  corrections  for  ordinates. 

106.  Calibration  of  a  Compensated  Wattmeter. — For  this 
purpose  a  potentiometer  may  be  used,  as  in  the  preceding 
method,  but  as  this  instrument  is  not  always  at  hand  a  method 
using  resistance  boxes  is  given  here.  The  principal  differ- 
ence between  the  compensated  wattmeter  and  the  uncompen- 
sated  one  is  that  the  former  measures  only  the  power  expended 
in  the  circuit  to  which  it  is  attached,  while  the  latter  measures, 
in  addition  to  this,  the  power  expended  in  its  own  shunt 
circuit. 

Thus  let  W,  Fig.   58,   represent  the  wattmeter  connected 


130 


ELECTRICAL  MEASUREMENTS 


in  the  circuit  to  measure  the  power  expended  in  H  and  C  to- 
gether. C  is  a  standard  resistance  for  use  in  the  measurement 
of  the  current,  and  H  is  sufficient  other  resistance  to  give 
the  required  amount  of  power.  As  the  power  expended  in 
the  shunt  coil,  a6,  is  not  measured  by  the  wattmeter,  it  should 


FIG.  58. — Calibration  of  the  Wattmeter,  W. 

not  be  measured  by  the  standard  cell;  therefore  C  is  placed  in- 
side next  to  H. 

The  calibration  circuit  consists  of  four  resistance  boxes, 
F,Q,  R,  and  S,  joined  in  series  with  a  battery  of  a  few  volts 
sufficient  to  maintain  a  small  constant  current  through  the 

!?  ; .,  T  n1S  drCUit  1S  j°ined  throuSh  the  galvanometer  to  the 

standard  cell  and  to  the  wattmeter  circuit  in  three  places  as 

iown,  each  connection  being  provided  with,  a  key      When 


MEASUREMENT  OF  POWER  131 

finally  balanced  no  current  flows  through  any  of  these  con- 
nections. 

The  measurements  are  made  as  follows.  First  P  is  set  at 
some  convenient  value,  say  1000  ohms,  and  some  of  the  remain- 
ing resistances  are  then  changed  until  there  is  no  deflection  of 
the  galvanometer  upon  closing  the  key  Ki.  This  fixes  the 
total  resistance  of  this  circuit  and  it  is  thereafter  kept  constant 
at  this  amount.  The  fall  of  potential  over  C  should  be  a  little 
larger  than  the  E.M.F.,  of  the  standard  cell.  It  will  then 
require  a  little  more  resistance  added  to  P  to  give  no  deflection 
when  the  key  K*  is  used.  This  is  added  by  varying  Q  and  S, 
keeping  their  sum  constant,  until  there  is  no  deflection  of  the 
galvanometer  upon  closing  the  key  Kz.  This  balance  measures 
the  value  of  the  current  through  C,  for 

1C  =  i(P  +  Q)  =  (P  +  Q)|, 

and  therefore, 

EP+Q 
~  C      P 

This  is  the  effective  current  actuating  the  wattmeter. 

The  fall  of  potential  over  CH  is  measured  in  the  same  way. 
This  will  be  greater  than  that  over  C  alone,  therefore  fora 
balance  it  will  require  a  resistance  greater  than  P  +  Q.  The 
resistance  R  is  now  varied,  keeping  R  +  S  constant,  until  there 
is  no  deflection  of  the  galvanometer  upon  closing  Ks.  The  fall 
of  potential  over  CH  is  then  the  same  as  that  over  the  three 
resistances,  P,  Q,  and  R.  That  is, 

V  =  i(P  +  Q  +  R)  =  (P  +  Q  +  fl)f  • 
The  power  measured  by  the  wattmeter  is,  then, 

w>       vr      E*(P  +  Q)(P  +  Q  +  R) 
P2C 

If  the  reading  of  the  wattmeter  is  W,  the  correction   to  be 
added  to  this  reading  is, 

c  ^  wf  -  W. 


132 


ELECTRICAL  MEASUREMENTS 


In  case  V  is  too  large  to  be  measured  directly  as  here  shown, 
there  may  be  placed  a  shunt,  AB,  around  H  as  shown  in  the 
preceding  method  and  the  potential  fall  over  A  alone  measured. 
The  total  fall  of  potential  over  H  is  then  readily  computed. 
The  addition  of  this  shunt  will  make  no  difference  in  the  watt- 
meter, since  H  with  its  shunt  now  replaces  H  alone  and  the 
wattmeter  measures  whatever  power  is  expended  in  either 
arrangement. 

106.  Calibration  of  a  High  Reading  Wattmeter. — A  high 
reading  wattmeter  is  one  that  measures  large  amounts  of 
power.  In  calibrating  such  a  wattmeter  it  is  often  impossible, 

and  usually  it  is  inconvenient,  to 
expend  sufficient  power  to  bring 
the  reading  up  to  the  high  values 
indicated  on  the  scale.  But  this  is 
not  necessary,  for  all  that  is  re- 
quired is  that  there  shall  be  a  large 
current  through  the  series  coil  and 
a  small  current  at  high  voltage 
through  the  shunt  circuit.  By  us- 
ing different  batteries  to  supply 
these  two  currents  there  need  be  no 
great  expenditure  of  energy.  As 
shown  in  the  figure,  the  battery 

B,  consisting  of  one  or  two  large  storage  battery  cells,  sup- 
plies a  large  current  through  the  series  coil  of  the  wattmeter 
and  the  low  standard  resistance  C.  The  latter  should  be  of 
such  a  value  that  the  fall  of  potential  over  it  will  be  a  volt  or 
less  in  order  that  this  may  be  readily  measured  by  the  poten- 
tiometer. Since  the  resistance  of  the  shunt  circuit  is  large 
it  will  require  a  large  number  of  cells  in  the  battery  B',  but  these 
cells  can  be  small  as  only  a  small  current  will  be  needed.  In 
parallel  with  this  circuit  is  placed  another  high  resistance  cir- 
cuit, AB,  so  divided  that  the  fall  of  potential  over  the  portion 
A  shall  be  about  one  volt.  The  calibration  is  then  the  same  as 
given  above  for  the  case  of  an  uncompensated  wattmeter,  and 


FIG. 


59. — Calibration  of  the 
Wattmeter,  W. 


MEASUREMENT  OF  POWER  133 

the  wattmeter  should  read  the  product  of  the  current  through 
C  and  the  voltage  across  A  and  B. 

If  the  reading  of  the  wattmeter  is  W,  the  correction  to  be 
added  to  this  reading  is 

c  =  V  I  -  W. 

In  making  this  calibration  it  is  best  to  have  the  currents 
through  the  wattmeter  about  the  same  as  will  be  used  when  the 
instrument  is  measuring  power. 


CHAPTER  IX 
MEASUREMENT  OF  CAPACITY 

107.  Laws  of  Condensers. — When  two  or  more  condensers 
are  joined  to  the  same  E.M.F.  each  one,  of  course,  becomes 
charged  to  this  difference  of  potential.  The  total  charge  is 

Q  =  Qi  +  Q2  =  CiE  +  CZE  =  CE, 

where  Qi  and  Q2  are  the  charges  in  each  condenser.  This 
combination  acts,  then,  like  a  single  condenser  whose  capacity 
is 

C  =  Ci  +  Cz 
Hence, 

Law  I. — The  combined  capacity  of  several  condensers  in  par- 
allel is  equal  to  the  sum  of  the  separate  capacities. 


FIG.  60. — Condensers  FIG.  61. — Condensers  in  series. 

in  parallel. 


If  the  condensers  are  joined  in  series  as  shown  in  Fig.  61  it  is 
evident  that  the  difference  of  potential  over  each  condenser  is 
only  a  part  of  the  total  E.M.F.  of  the  battery.  The  amount  of 
charge  in  each  of  the  condensers  will  be  the  same,  for,  being 
joined  in  series,  whatever  electricity  leaves  one  must  go  into 
the  other,  if  the  intermediate  parts  are  well  insulated. 

134 


MEASUREMENT  OF  CAPACITY  135 

The  potential  differences  across  each  condenser,  respectively 
will  be, 


r  3    --    K  4  —   p 
Cs 

Adding  these  equations  gives 

.0 

'C 

Hence  the  equivalent  capacity,  C,  of  the  combination  is 
given  by  the  relation 

1-1+1+1 
n         n     *    n     '    n 

O  Ol  1/2  ^3 

Law  77. — When  several  condenser^  are  connected  in  series,  the 
joint  capacity  is  the  reciprocal  of  the  sum  of  the  reciprocals  of 
the  several  capacities. 

This  is  similar  to  the  law  for  resistances  in  parallel. 

108.    Comparison  of  Capacities  by  Direct  Deflection. — When 
a  condenser  of  capacity  C  farads  is  charged 
to   a  potential  difference  of  E  volts,  th  e     j         V£/        1 
quantity  it  contains  is 


Q  =  CE,  coulombs.  (1) 

If  this  quantity  is  discharged  through  a 
ballistic  galvanometer,  giving  a  fling  of  d  mm.  as  measured  on 
the  scale,  we  have 

Q  =  fed.  (2) 

Combining  (1)  and  (2)  gives 

C  =  4<*  (3)  • 


136 


ELECTRICAL  MEASUREMENTS 


Now  if  the  same  experiment  is  repeated  using  the  same 
battery  and  galvanometer,  but  with  another  condenser  of 
capacity  Ci  we  have 

^ 

r  •-     j  fd.\ 

O  i  -  ™  di  (4; 


and  from  (3) 


(5) 


If  C  is  a  known  capacity  then  the  value  of  Ci  can  be  deter- 
mined as  exactly  as  the  flings  d  and  di  can  be  measured. 
Each  of  these  should  be  taken  several  times  and  the  mean  values 
used  in  the  computation. 

109.  Bridge  Method  for  Comparing  Two  Capacities. — This  is  a 
null  method  and  therefore  capable  of  more  exact  measure- 
ments than  the  preceding.  The  two  condensers  are  placed  in 

two  arms  of  a  Wheatstone  bridge 
setup  as  shown  in  Fig.  63. 

When  the  key  is  depressed  both 
condensers  become  charged  to  the 
full  potential  difference  of  E,  and 
the  points  A,  B  and  C  all  come  to 
the  same  potential.  If  no  charge 
passes  through  the  galvanometer 
then  Ci  is  charged  through  RI  and 
Cz  through  R%.  During  the  very  short  interval  that  is  re- 
quired to  charge  the  condensers  there  will  be  transient  currents 
through  RI  and  Rz,  and  perhaps  in  the  galvanometer  also. 
By  working  the  charge  and  discharge  key  quickly  the  de- 
flection of  the  galvanometer  may  be  checked  even  when  RI  and 
Rz  are  far  from  a  balance.  The  galvanometer  should  be  thus 
protected. 

Applying  KirchhofTs  second  law  to  the  circuit  B  A  C  B,  &t 
any  instant  while  the  condensers  are  being  charged  gives, 

D  •    i    r  dii      p  .        r  ^2  _i_  f '  -I-  7"  —  —  0 

11         1  dt         2  2         2  dt  dt 


FIG.  63. — Bridge  method. 


MEASUREMENT  OF  CAPACITY  137 

where  Li  -77,  etc.,  are  the  E.M.F's.  due  to  self  induction  in  the 

corresponding  branches. 

Multiplying  this  equation  by  dt  and  integrating  it  from 
t  =  0,  to  t  =  T,  for  the  time  limits,  and  between  the  corre- 
sponding limits  i  =  0  and  i  =  0  for  the  currents  gives 


It  is  to  be  noticed  that  no  deflection  of  the  galvanometer 
does  not  mean  no  current  through  it,  as  there  may  be  currents 
through  it  in  both  directions  before  the  condensers  are  fully 
charged.  But  zero  deflection  means  that  as  much  electricity 
has  passed  through  the  galvanometer  in  one  direction  as  in  the 
other,  and  that,  taking  account  of  signs,  the  total  quantity 
through  the  galvanometer  has  been  zero.  Hence,  when  RI 
and  R%  are  adjusted  so  that  opening  or  closing  K  produces  no 
deflection  of  the  galvanometer,  q  =  0. 
Then 

=  #222  (A) 


But  since  no  charge  passed  through  the  galvanometer, 

qi  =  CiE        and        #2 
Hence,  substituting  in  (A),  gives 


The  resistances  RI  and  R2  should  be  large,  5000  ohms  or 
more,  so  that  the  fall  of  potential  produced  by  the  small 
charging  currents  may  be  appreciable.  While  self  inductance 
in  these  arms  does  not  affect  the  result,  as  seen  by  the  result 
of  the  preceding  integration,  a  large  amount  may  render  it 
more  difficult  to  determine  when  a  balance  has  been  reached. 

Make  five  determinations  of  each  unknown  capacity  by 
using  various  values  for  RI  and  finding  the  corresponding 
values  of  R^.  Check  results  by  measuring  the  capacity  of 
the  condensers  when  joined  in  series  and  in  parallel. 


138  ELECTRICAL  MEASUREMENTS 

The  data  can  be  recorded  in  a  form  like  the  following: 


fit 

fij 

C2 

Ci 

110.  Comparison  of  Capacities  by  Gott's  Method. — This  is 
another  bridge  method  and  differs  in  arrangement  from  the 
preceding  only  by  having  the  galvanometer  and  battery 
interchanged.  In  the  bridge  method  the  balance  is  obtained 
for  the  conditions  which  exist  during  the  charging,  or  the  dis- 
charging, of  the  condensers.  In  the  present  method  the  capa- 
cities of  the  condensers  are  compared  after  everything  has 

reached  the  steady  and  permanent 
condition. 

The  arrangement  is  shown  in  Fig. 
64.  When  the  battery  key,  K' ',  is 
closed  the  two  condensers  in  series 
are  charged  to  the  difference  of 
potential  between  A  and  D.  The 
point  B,  between  the  two  con- 
densers, has  a  potential  interme- 
diate between  that  of  A  and  D. 

The  measurement  consists  in  adjusting  Ri  and  Rz  until  C  on 
the  upper  circuit,  has  the  same  potential  as  B  in  precisely  the 
same  way  as  in  the  measurement  of  resistance  by  the  Wheat- 
stone  bridge  method.  When  this  adjustment  is  correct,  there 
will  be  no  deflection  of  the  galvanometer  upon  closing  K". 

Since  the  condensers  are  joined  in  series,  each  one  must 
contain  the  same  charge.  The  point  B  is  insulated  as  long  as 
K"  remains  open,  and  therefore  whatever  charge  appears  on 
one  condenser  must  have  come  from  the  other  one.  Moreover, 
for  a  balance  the  closing  of  K"  produces  no  deflection  of  the 


FIG.  64.— Gott's  method. 


MEASUREMENT  OF  CAPACITY  139 

galvanometer,  i.e.,  there  is  no  change  in  the  charges  on  the 
conductors  joined  at  B. 

Applying  Kirchhoff's  second  law  to  the  circuits  BACB  and 
DBCD  gives, 


0         and  -  -  R2i  =  0 

Ci  02 

from  which, 

Ti    -  To— 

L/l    —    O2r>   * 
til 

It  should  be  noted  that  closing  K"  brings  B  to  the  potential 
of  C  whether  there  is  a  balance  or  not.  A  second  closing  of  K" 
can  produce  no  deflection.  It  is  necessary  therefore  to  com- 
pletely discharge  the  condensers  after  each  closing  of  K"  . 
This  can  be  most  quickly  done  by  opening  K'  before  open- 
ing K". 

If  one  of  the  condensers  has  considerable  absorption  or 
leakage  it  will  seriously  influence  the  results,  for  after  some 
of  the  charge  has  leaked  away  it  is  no  longer  true  to  say  that 
the  charges  in  the  two  condensers  are  equal.  The  effect  of  this 
source  of  error  is  reduced  by  closing  K"  as  soon  as  possible 
after  closing  K'.  Sometimes  a  rapidly  alternating  E.M.F.  is 
used  in  place  of  the  battery  and  key,  and  a  telephone  receiver 
instead  of  the  galvanometer  and  key.  This  would  only  be 
allowable  when  the  resistances  R  i  and  Rz  are  free  from  self 
induction  and  capacity. 

If  this  is  true,  it  is  also  allowable  to  omit  K"  from  Fig.  64 
and  obsrve  the  deflect;ons  of  the  galvanometer  when  K'  is 
closed  or  opened. 

The  data  can  be  recorded  in  the  same  form  as  used  for  the 
preceding  experiment. 

111.  Comparison  of  Capacities  by  the  Method  of  Mixtures. 
—  This  method  was  devised  by  Lord  Kelvin  to  avoid  some  of 
the  difficulties  in  the  preceding  methods.  It  is  especially 
applicable  to  cases  where  the  two  capacities  are  very  dissimilar, 
e.g.,  if  the  capacity  of  a  long  cable  is  to  be  compared  with  that 


140 


ELECTRICAL  MEASUREMENTS 


of  a  standard  condenser.  The  method  consists  of  charging 
the  condensers  to  such  potentials  that  each  will  contain  the 
same  quantity.  They  are  then  discharged,  the  one  into  the 
other,  and  the  charges  allowed  to  mix.  If  the  charges  are  not 
equal,  the  difference  will  remain  in  the  condensers  and  is 
later  discharged  through  the  galvanometer. 

The  arrangements  and  connections  are  shown  in  Fig.  65. 
Two  moderately  high  resistance  boxes,  RI  and  R2  are  joined 
in  series  with  a  battery  of  a  few  cells.  In  parallel  with  each 


fti 


Jfe 


FIG.  65. — Method  of  mixtures. 


resistance  is  joined  one  of  the  capacities  to  be  compared, 
the  connection  being  made  through  the  double  switch,  S. 
Each  condenser  is  thus  charged  to  the  potential  difference 
over  the  corresponding  resistance.  The  charges  are,  then, 


Qi  = 


and 


where  i  denotes  the  steady  current  through  R  i  and  R^. 

By  throwing  the  switch  the  other  way  the  condensers  are 
joined  together  and  the  charges  mix,  the  difference,  if  any, 
being  discharged  through  the  galvanometer  by  the  key, 
K.  No  deflection  indicates  that  the  charges  are  equal,  and 


MEASUREMENT  OF  CAPACITY 


141 


RI  and  RI  are  adjusted  until  this  balance  is  obtained.     Then, 

f\         f\  *-vwi         /"I    ~D   /*     «_    f^    ~D     * 

Vi  —  V2     or     Uin/ifr  —  \JZL\I%I, 
from  which 


Ki 


In  order  to  avoid  as  far  as  possible  the  effects  of  leakage  and 
absorption  the  keys  should  be  worked  as  quickly  and  uni- 
formly as  possible.  This  can  best  be  done  by  using  the 
special  testing  key,  shown  in  principle  in  Fig.  66.  This  is  a 
regular  Wheatstone  bridge  key  with  the  addition  of  a  break 
contact  on  the  top  of  each  blade.  The  top  and  bottom  blades  of 
the  key  are  both  joined  to  one  condenser,  say  Ci,  and  the  mid- 
dle blade  is  joined  to  the 
other  condenser,  C2,  as  indi- 
cated in  the  figure.  The  con- 
densers are  thus  connected 
to  the  resistances  through 
the  contacts  R  i  and  R%  re- 
spectively. The  key  is  pro- 
vided with  an  extra  insula-  FIG.  66. — Triple  make-and-break  key. 
tion  block  to  prevent  the  up- 
per blades  from  coming  in  contact  when  the  key  is  depressed. 
Also  the  usual  insulation  between  the  lower  blades  can 
be  moved  to  one  side,  thus  allowing  these  blades  to  come  in 
contact. 

When  the  key  is  depressed,  the  first  action  is  to  disconnect 
the  condensers  Ci  and  C2  from  the  charging  contacts,  RI 
and  Rz.  When  the  key  is  further  depressed  the  two  lower 
blades  come  in  contact,  thus  joining  the  condensers  together 
and  mixing  their  charges.  Finally,  when  the  lower  blade 
touches  the  last  contact,  G,  the  charge  remaining  in  the  con- 
densers after  the  mixing,  is  discharged  through  the  galva- 
nometer. All  of  the  operations  for  this  experiment  are  thus 
performed  while  the  key  is  being  depressed. 

The  data  form  in  Article  109  may  be  used  here  also. 


142 


ELECTRICAL  MEASUREMENTS 


112.  Study  of  Residual  Discharges. — When  a  condenser 
or  a  cable  is  charged  for  a  long  time  and  then  discharged  it  is 
nearly  always  found  that  the  quantity  of  electricity  obtained 
from  the  condenser  on  discharge  is  less  than  the  total  amount  of 
the  original  charge.  The  remainder  of  the  original  charge  is 
said  to  be  "  absorbed,"  meaning  thereby  that  this  charge  re- 
mains in  the  condenser  after  the  plates  have  been  brought 
to  the  same  potential,  but  not  specifying  the  manner  in  which 
it  is  retained.  After  a  short  interval  of  time  a  portion  of  this 

absorbed  charge  is  released  and  can 
be  discharged  by  again  joining  the 
two  plates  of  the  condenser;  and 
this  is  termed  a  residual  discharge. 
Several  such  residual  discharges  can 
be  obtained  from  ordinary  con- 
densers and  a  great  many  from  a 
poor  condenser. 

This  phenomenon  depends  upon 
the  material  composing  the  dielec- 
tric between  the  plates  of  the  con- 
denser, and  is  the  more  marked  the 
greater  the  heterogeneity  of  the  di- 
electric. In  case  the  dielectric  is 
air,  or  a  sheet  cut  from  a  crystal  of 
quartz  or  iceland  spar  (i.e.,  a  homogeneous  substance)  there  is 
no  residual  discharge.  The  charge  which  is  thus  absorbed 
and  can  be  recovered  later  should  not  be  confused  with  any 
leakage  there  may  be  through  the  condenser. 

To  make  a  short  study  of  residual  discharges  proceed  as 
follows:  Charge  a  paraffined  paper  condenser  from  six  or 
eight  cells  for  3  minutes.  When  ready  to  begin  observations 
on  residual  discharges  disconnect  the  battery  from  the 
condenser  and  thoroughly  discharge  the  latter  by  throwing 
the  switch  S  to  the  other  side  for  5  seconds  and  then  leaving  it 
open.  Wait  1  minute  and  again  discharge  the  condenser — 
this  time  through  the  galvanometer  and  observe  the  deflection. 


FIG.  67. 


MEASUREMENT  OF  CAPACITY  143 

The  key  K  is  supposed  to  remain  closed  on  the  lower  contact 
shown  in  Fig.  67.  Probably  this  would  be  the  upper  contact 
on  the  actual  key.  Pressing  the  key  to  the  other  contact  joint 
the  condenser  to  the  galvanometer.  Discharge  through 
the  galvanometer  at  minute  intervals  for  half  an  hour. 
Express  the  results  in  the  form  of  a  curve  taking  time  for  ab- 
scissae and  for  ordinates  the  sum  of  all  previous  discharges. 
In  some  instances  the  sum  of  the  residual  discharges  is  greater 
than  the  original  discharge.  Repeat,  charging  the  condenser 
for  15  minutes  in  order  to  observe  the  effect  of  the  time 
of  charging  upon  the  amount  of  the  residual  charge. 

113.  The  Ballistic  Galvanometer. — A  ballistic  galvanometer 
is  one  designed  to  measure  transient  currents,  or  quantities  of 
electricity,  like  the  induced  current  in  a  secondary  coil  or  the 
discharge  of  a  condenser.     The  duration  of  the  current  is  very 
brief  as  compared  with  the  period  of  the  needle  of  the  galvan- 
ometer, and  thus  the  coil  remains  practically  at  rest  until  the 
entire   quantity   has   passed   through   it.     It   is   impossible, 
therefore,  to  obtain  a  steady  deflection,  but  the  galvanometer 
coil  gives  a  sudden  throw  and  then  settles  down  to  the  position 
of  rest.     The  extent  of  this  throw  gives  a  measure  of  the 
quantity  of  electricity  that  has  passed   through  the  galvan- 
ometer: the  greater  the  quantity  the  greater  the  throw,  the 
exact   relation   depending   upon   the   type   of  galvanometer 
employed. 

These  galvanometers  are  of  two  general  types:  (1)  The 
magnet  may  be  fixed,  and  the  coil  suspended  and  movable; 
or,  (2)  the  magnet  may  be  suspended  and  movable  and  the  coil 
fixed. 

114.  Turning  Moment  Due  to  a  Current. — In  galvanometers 
of  the  first  type  the  poles  are  usually  so  shaped  as  to  give  a 
radial  magnetic  field.     Then  the  angular  twist  of  the  suspen- 
sion produced  by  the  couple  due  to  a  steady  current  flowing 
through  the  coil,  is  directly  proportional  to  the  current,  and  is 
independent  of  any  particular  position  of  the  coil.     If  /  is  the 
value  of  the  steady  current,  and  G  the  torque  per  unit  current 


144  ELECTRICAL  MEASUREMENTS 

(a  constant  of  the  galvanometer),  then  GI  is  the  moment  of  the 
forces  due  to  the  steady  current  /  tending  to  turn  the  coil. 
When  a  condition  of  equilibrium  is  reached,  we  have,  by  equat- 
ing moments, 

GI  =  a<f>  (1) 

where  $  is  the  angle  through  which  the  coil  has  been  turned, 
and  a  is  the  torque  of  the  suspension,  or  the  couple  exerted  by 
the  suspension  when  the  coil  is  turned  through  unit  angle. 

When  the  current  is  not  steady,  but  is  variable,  there  can 
be  no  steady  deflection  $,  but  an  angular  acceleration  is  given 
to  the  coil,  the  amount  of  which  is  measured  by  the  subsequent 
fling.  If  i  is  the  value  of  the  transient  current  at  any  instant, 
then  Gi  will  be  the  moment  of  the  forces  due  to  this  current, 
and  the  equation  of  motion  of  the  coil  is, 


But  </>  =  0,  before  the  suspension  is  twisted,  and  therefore  at 
the  start, 

«-*$,'  .  (2) 

where  K  is  the  moment  of  inertia  of  the  moving  system  (coil, 
mirror,  etc.),  and  -^-  the  angular  acceleration,  where  o>  is  the 

angular  velocity. 
From  (2)  we  have  the  relation  that, 

Gidi  =  Kdw  (3) 

Now  the  integral  of  idt  is  simply  Q,the  total  quantity  of  electricity 
that  has  passed  through  the  galvanometer.    Therefore,  from  (3)  , 


(4) 
and  it  remains  to  measure  co. 

115.  To  Express  o>  in  Terms  of  Known  Quantities.  —  The 
kinetic  energy  of  a  system  of  moment  of  inertia  K  and  rotating 
with  this  angular  velocity  o>,  is 


K.E.  =    #co2  (5) 


MEASUREMENT  OF  CAPACITY  145 

and  the  coil  will  continue  to  turn  until  this  energy  has  all  been 
expended  in  twisting  the  suspension.  The  suspension  thus 
gains  potential  energy,  and  when  the  coil  comes  to  rest  (for 
an  instant  at  the  position  of  its  maximum  fling),  the  poten- 
tial energy  of  the  twisted  suspension  equals  the  kinetic  en- 
ergy with  which  the  coil  started  —  save  for  a  slight  amount 
lost  through  friction,  air  currents,  etc. 

The  amount  of  potential  energy  thus  gained  by  the  suspen- 
sion can  be  computed  independently  by  considering  the  work 
that  has  been  done  upon  it.  The  torque  exerted  by  the  sus- 
pension when  twisted  an  angle  $  is  a<j>;  and  the  work  to  twist  it 
through  the  further  angle  d(j>  is  a<j>d<i>.  The  total  work  done  in 
twisting  the  suspension  from  zero  to  an  angle  6  is, 


r 

I 

Jo 


o 

and  this  is  equal  to  the  potential  energy  gained  by  the  suspen- 
sion. 

Equating  the  kinetic  energy  at  the  beginning  of  the  swing 
to  the  potential  energy  at  the  end,  we  have  the  relation  that, 


where  0  is  the  maximum  throw. 
Solving  for  w  gives, 


and  substituting  this  value  in  (4)  gives, 


From  equation  (1)  we  have  the  relation  that 

°L  -L 
G~  <f> 

and  putting  this  value  into  the  expression  for  Q  gives, 


10 


146  ELECTRICAL  MEASUREMENTS 

1?- 

The  value  of  —  can  be  determined  by  finding  the  period  of 

oscillation  of  the  moving  system. 

Usually  the  deflections,  d'  and  d,  as  read  from  the  scale,  are 
proportional  to  the  angles  0  and  6.     If  they  are  not  they  can  be 

corrected  by  means  of  a  calibration  curve.     Writing,  then,  —f 
for  —  ,  gives 


where  F  is  the  figure  of  merit  of  the  galvanometer.1 

116.  Period  of  Oscillation.  —  In  a  galvanometer  of  the  mov- 
ing coil  type  the  restoring  force  is  entirely  due  to  the  suspen- 
sion, and  therefore  varies  directly  as  the  angle  of  twist,  0. 
The  equation  of  motion  is  then, 


where  0  is  the  angular  displacement,  and  K  and  a  have  the 
same  meanings  as  above. 

We  find 2  the  solution  of  this  equation  to  be  the  simple  har- 
monic function, 

0  =  c  sin    +1—  t  H- 

where  c  and  /  are  the  two  constants  of  integration.  At  the 
end  of  the  first  period  the  value  of  0  will  return  to  this  same 
value,  but  the  angle  part  of  the  above  expression  will  have 
increased.  This  increase  at  the  end  of  the  first  period  will  be 
27r,  and  the  angle  will  now  be 


If  we  think  of  the  angle  as  increasing  because  the  value  of  t 

1  See  Article  40 

2  Murray,  Differential  Equations,  p.  97. 


MEASUREMENT  OF  CAPACITY  147 

has  increased,  then  at  the  time  t  +  T}  corresponding  to  the 
end  of  the  first  period,  the  value  of  the  angle  will  be 


Since  the  angle  is  the  same  in  either  case,  these  two  values  are 
equal,  and  we  have 


TT          '      J      ' 

Solving  for  T  we  have 

r 

"  a 
from  which 

IK        T 

a 

Substituting  this  value  in  equation  (7), 

FT  T 

9-9r 

Thus  our  final  equation  contains  only  measurable  values. 

But  this  equation  does  not  take  into  account  the  effect  of 
damping  on  the  reading  of  the  galvanometer.  If  this  is  not  too 
great,  a  correction  can  be  made  for  it,  as  follows. 

117.  Correction  for  Damping. — Let  0i,  02,  0s,  etc.,  be  the 
successive  deflections  of  the  galvanometer  to  the  right  and  left 
when  it  is  allowed  to  swing  freely  after  the  discharge  of  the 
condenser  through  it.  It  is  observed  that  each  deflection  is 
less  than  the  one  before  it  by  a  certain  constant  ratio,  so  that 


If  these  deflections  are  laid  off  as  ordinates,  each  one  being 
erected  at  that  point  on  the  axis  of  abscissas  corresponding 
to  the  time  at  which  it  was  observed,  they  would  appear  as  in 


148 


ELECTRICAL  MEASUREMENTS 


Fig.  68.  Time  is  reckoned  from  the  discharge  of  the  condenser 
and  when  the  galvanometer  coil  begins  to  move.  After  half 
of  a  single  period  the  first  throw,  0i,  is  observed.  The  succeed- 
ing deflections  follow  at  equal  intervals  of  a  whole  single 
period.  Knowing  the  value  of  /,  any  given  deflection,  say  0i, 
can  be  computed  from  the  later  readings,  since 

0i  =  02/  =  0s/2  =  04/3  =  etc. 

where  the  exponent  of  /  in  each  case  is  equal  to  the  number  of 
single  periods  between  the  desired  and  observed  deflections. 


Q 


Time 

FIG.  68. — Damping  of  a  galvanometer. 

Had  there  been  no  damping,  all  of  these  deflections  would 
have  been  larger,  and  each  equal  to  0,  the  value  of  which  is 

/I ,      ft      J*  2 

where  the  exponent  of  /  is  5  because  the  interval  between  0 

and  0i  is  half  of  a  single  period. 
Since 

..   01    02    _    01  ~h   02    $1 

02  03  02     I     03  ^2 

the  value  of/  is  seen  to  be  also  given  by  the  ratio  of  two  con- 
secutive swings — a  swing  being  the  full  amplitude  from  the 
turning  point  at  the  right  to  the  turning  point  at  the  left. 


MEASUREMENT  OF  CAPACITY  149 

Using  this  value  of  /  gives, 

— .©»• 

118.  Final   Formulas. — The   final    corrected   value  for   Q 
becomes,  then, 

TTim  /c*  v    1 

h=  kdi  (8) 


Where  k  represents  all  the  constants  in  this  equation,  and  d\ 
is  the  observed  value  of  the  first  deflection. 

119.  Absolute  Capacity  of  a  Condenser. — By  "absolute 
capacity"  is  meant  the  value  of  the  capacity  of  a  condenser 
determined  independently  of  any  other  capacity.  The 
comparison  of  two  condensers  can 
only  give  their  relative  capacities. 

The  fundamental' relation  for  a 
condenser  is 


£ 


f 

1      * 

1 

c 

—  I  s  [~ 

p  \  — 

Fio.  69. 

Q  =  CE. 

If  then  by  means  of  a  ballistic  gal- 
vanometer we  can  measure  the 
quantity  Q  which  will  charge  a  con- 
denser to  the  potential  difference 
E,  its  capacity  can  be  determined 

from  this  relation.  The  setup  is  shown  in  the  figure.  R  is 
a  high  resistance  of  about  100,000  ohms,  and  together  with 
P  and  S  forms  an  arrangment  for  finding  the  figure  of  merit 
of  a  galvanometer.  By  closing  K"  a  steady  current  can  be 
passed  through  the  galvanometer,  and  the  resulting  steady 
deflection  d'  observed.  If  the  galvanometer  is  a  sensitive  one 
it  will  be  necessary  to  use  only  a  few  ohms  in  P  while  S 
should  be  several  thousand  ohms.  The  current  through  the 
galvanometer  is: 


P  +  S    R+g 


=  Fd'. 


150  ELECTRICAL  MEASUREMENTS 

The  condenser  may  be  joined  in  as  shown  where  it  is  charged 
from  the  potential  across  the  resistance  S.  In  case  this  charge 
is  too  great  the  fall  of  potential  across  S  can  be  reduced  by 
inserting  another  resistance  in  series  with  the  battery.  (It  is 
best  not  to  use  a  shunt  with  the  galvanometer,  unless  it  be  for 
damping.) 

The  quantity  of  electricity  passing  through  the  galvanometer 
has  been  found  to  be  given  by  the  expression, 

Q=^ 


where  the  symbols  have  the  same  meanings  as  before.  Each 
quantity  can  be  determined,  and  therefore  the  charge  in  the 
condenser  can  be  found. 

The  difference  of  potential,  E,  to  which  the  condenser  is 
charged  is, 


where  V  denotes  the  fall  of  potential  over  both  P  and  S.     We 
thus  have  for  the  capacity  of  the  condenser, 

c          p 


E  ~  2*  S(R  +  g)    S       d 


CHAPTER  X 


THE  MAGNETIC  CIRCUIT 

120.  Introduction. — It  has  been  already  noticed  that  an 
electric  current  produces  a  magnetic  field,  and  this  phenomenon 
is  especially  striking  when  the  current  is  not  steady.     In  the 
case  of  the  dynamo  and  the  transformer  the  current  is  gener- 
ated by  the  varying  magnetic  field.     It  is  the  object  of  this 
chapter  to  inquire  more  minutely  into  this  intimate  relation 
between  the  electric  current  and  the 

magnetic  field  with  which  it  is  asso- 
ciated. 

In  order  to  fix  ideas  by  a  con- 
crete example  let  us  consider  a  sim- 
ple dynamo  as  outlined  in  Fig.  70. 
A  is  the  armature  which  rotates  be- 
tween the  pole  pieces,  PP.  FF  are 
the  fields,  which  are  connected 
above  by  the  yoke,  Y.  The  mag-  FIG.  70.— A  magnetic  circuit, 
netism  is  produced  by  an  electric 

current  through  the  many  turns  of  the  field  coils.  The  closed 
path  APFYFPA  is  called  the  magnetic  circuit,  and  in  this 
case  it  is  almost  entirely  of  iron,  because  iron  is  one  of  the 
best  conductors  of  the  magnetic  flux. 

What  is  this  relation  between  electricity  and  magnetism? 
How  is  the  magnetism  of  the  dynamo  produced?  Why  is  so 
much  power  required  to  run  it?  In  the  pages  that  follow  we 
will  endeavor  to  find  the  answers  to  these  questions,  and  to 
learn  some  things  regarding  the  nature  of  magnetism. 

121.  The    magnetic    circuit    is   analogous   to   the   electric 
circuit,  but  with  one  very  important  difference.     As  there  are 

151 


/' 

-y  

**x 

1 

I       '  

IpL 

—  L'p~ 

_^  — 

.  L_^ 

V 

p  ^-- 

BE 

-'  p 

152  ELECTRICAL  MEASUREMENTS 

no  insulators  for  magnetic  flux  it  is  not  possible  to  confine  the 
magnetic  flux  to  any  chosen  path  as  in  the  electrical  analogy. 
It  is  as  though  the  electric  circuit  were  immersed  in  a  con- 
ducting bath  of  salt  water  where  the  greater  part  of  the  current 
would  follow  the  best  conductor  but  some  portion  of  it  would 
flow  through  the  surrounding  medium.  The  law  of  the 
magnetic  circuit  is  precisely  similar  to  Ohm's  law  under  these 
conditions,  and  can  be  written 

Magneto-motive  Force  ,., 

Magnetic  Flux  =  -         p  ,  (1) 

Reluctance 

This  law  can  be  applied  to  any  magnetic  circuit,  and  it 
holds  true  for  circuits  of  any  material,  with  one  exception. 
For  substances  which  show  magnetic  hysteresis — such  as  iron, 
steel,  nickel,  etc. — this  law  may  be  applied  only  when  the 
substance  is  being  magnetized  by  an  increasing  M.M.F.  which 
is  greater  than  any  M.M.F.  that  has  been  used  since  the  sub- 
stance was  unmagnetized. 

122.  Magnetic  flux  thus  has  its  analogy  in  electric  current. 
It  is  always  continuous  and  forms  a  closed  circuit  with  itself. 
At  one  part  it  may  be  spread  out  over  a  large  area  and  at 
another  part  of  the  circuit  be  confined  within  narrow  limits, 
but  the  total  amount  of  magnetic  flux  is  the  same  for  every 
cross  section  of  the  entire  circuit.  When  once  the  magnetic 
flux  has  been  established  no  further  expenditure  of  energy 
is  required  to  maintain  it.  In  this  respect  it  is  quite  different 
from  the  electric  current. 

The  direction  of  the  magnetic  flux  can  be  represented  by 
" stream  lines"  in  space,  these  lines  being  closer  together 
where  the  flux  is  more  intense  and  farther  apart  where  the  flux 
spreads  out  over  a  wider  path.  Since  the  flux  is  continuous, 
each  line  forms  a  closed  curve,  and  two  lines  can  never  inter- 
sect or  cross  one  another.  There  is  no  limit  to  the  number  of 
such  stream  lines  that  may  be  drawn,  but  by  convention  we 
consider  only  as  many  as  are  numerically  equal  to  the  value  of 
the  flux  as  given  by  equation  (1). 


THE  MAGNETIC  CIRCUIT  153 

The  unit  of  flux  is  called  a  maxwell,  and  is  represented,  then, 
by  one  line.  These  lines  are  sometimes  called  "  lines  of  induc- 
tion." It  will  be  shown  later  that  a  magnetic  flux  is  measured 
by  the  E.M.F.  induced  in  a  conductor  which  cuts  across  the 
flux,  and  this  effect  gives  a  basis  for  defining  the  value  of  a 
maxwell.  Looking  forward  to  Article  139,  there  is  found  the 
following  statement. 

A  maxwell  is  the  amount  of  magnetic  flux  cut  each  second  by  a 
conductor  in  which  there  is  induced  an  E.M.F.  of  one  C.G.S.  unit. 

The  name  " maxwell"  for  unit  magnetic  flux  was  adopted  by 
the  International  Electrical  Congress  at  Paris  in  1900.  * 

If  a  ballistic  galvanometer  is  connected  to  the  conductor 
when  it  is  cutting  the  magnetic  flux  there  will  be  a  deflection 
of  the  galvanometer,  and  this  deflection  may  be  taken  as  a 
measure  of  the  total  flux  cut  by  the  conductor.  It  does  not 
matter  whether  the  conductor  is  moved,  or  whether  the  flux 
moves  across  the  stationary  conductor,  the  effect  on  the  gal- 
vanometer is  the  same. 

If  then,  for  example,  it  is  desired  to  measure  the  flux  through 
a  bar  magnet,  a  turn  of  wire  is  placed  around  the  bar  and  its 
ends  joined  to  a  ballistic  galvanometer.  When  the  magnet 
is  suddenly  withdrawn  from  the  loop  of  wire  there  is  a  deflec- 
tion of  the  galvanometer  which  is  proportional  to  the  total 
amount  of  flux  cut  across  the  wire.  Since  in  the  first  position 
the  loop  of  wire  enclosed  all  of  the  flux  passing  through  the 
magnet,  and  after  withdrawing  the  magnet  the  wire  encloses 
none  of  it,  ail  of  the  flux  must  have  cut  across  the  wire  at  some 
time  during  the  withdrawal  of  the  magnet. 

If  the  coil  around  the  magnet  consisted  of  more  than  one 
turn  of  wire,  the  throw  of  the  galvanometer  would  have  been 
correspondingly  greater.  In  other  words,  the  galvanometer 
measures  the  "flux  turns,"  i.e.,  the  actual  flux  multiplied  by 
the  number  of  turns  of  wire  across  which  it  is  cut. 

123.  Magnetomotive  force  is  the  cause  of  magnetic  flux. 
It  may  be  due  to  electric  currents  or  to  permanent  magnets. 
Like  its  analog,  E.M.F. ,  which  is  measured  by  the  work  per 

<  Electrical  Review,  vol.  47,  p.  441,  1900. 


154  ELECTRICAL  MEASUREMENTS 

unit  charge  required  to  carry  a  quantity  of  electricity  once 
around  the  electric  circuit — magnetomotive  force  is  measured, 
in  precisely  the  same  way,  by  the  work  per  unit  pole  required 
to  carry  a  magnetic  pole  once  around  the  magnetic  circuit. 
This  unit  of  M.M.F.  is  often  called  a  gilbert.1 

Definition. — A  gilbert  is  the  magnetomotive  force  in  a  circuit 
when  one  erg  per  unit  pole  is  required  to  carry  a  magnetic  pole 
around  the  circuit. 

It  is  not  necessary  to  actually  take  the  unit  pole  around 
the  circuit,  for  if  we  know  enough  about  the  circuit  it  will  be 
possible  to  compute  the  number  of  ergs  of  work  that  would  be 
done  in  carrying  the  pole  along  the  given  path,  and  thus  to 
determine  the  value  of  the  M.M.F.  This  computation  is 
especially  simple  when  the  magnetic  forces  are  due  solely  to 
electric  currents,  for  from  the  definition  of  unit  current  the 
work  to  carry  a  unit  pole 'once  around  any  closed  path  is 
4ir  times  the  current  enclosed  by  the  path.  Therefore  the 
magnetomotive  force  in  the  complete  circuit  is, 

M.M.F.  =  OAir  NI  gilberts, 

where  I  is  the  current  in  amperes  and  N  is  the  number  of  times 
this  current  is  linked  with  the  magnetic  circuit.  The  product 
NI  is  called  " ampere  turns"  and  differs  from  M.M.F.  only  by 
the  constant  factor  of  0.47T  =  1.26.  It  is  evident  that  a  small 
current  in  a  coil  of  many  turns  will  produce  the  same  magneto- 
motive force  as  a  large  current  in  a  coil  of  few  turns. 

124.  Reluctance,  as  magnetic  resistance  is  called,  corre- 
sponds to  electrical  resistance,  and  it  depends  in  the  same  way 
upon  the  dimensions  of  the  circuit  and  the  material  of  which 
it  is  composed.  Iron  in  its  various  forms  is  one  of  the  best 
conductors  of  magnetic  flux.  Most  other  substances  are 
rather  poor  conductors,  but  there  are  no  magnetic  insulators. 

Although  the  laws  of  the  electric  and  the  magnetic  circuits 
are  similar  in  many  respects,  it  must  not  be  supposed  that  the 
analogies  hold  throughout.  It  requires  no  energy  to  maintain 
the  magnetic  flux  when  once  established.  There  is  therefore 

1  Sometimes  a  slightly  larger  unit  of  one  ampere  turn  is  used. 


THE  MAGNETIC  CIRCUIT  155 

no  analogy  to  Joule's  law  for  the  energy  continually  being 
dissipated  in  heat  during  the  flow  of  current.  There  is  a 
difference,  too,  between  electric  resistance  and  magnetic 
reluctance  inasmuch  as  the  former  is  constant  for  all  ranges 
of  current,  while  the  value  of  the  reluctance  of  a  magnetic 
circuit  depends  upon  the  value  of  the  magnetic  flux. 

If  two  paths  are  open  to  the  magnetic  flux  it  will  divide  just 
as  an  electric  current  would  do  between  two  wires  in  parallel. 
In  a  circuit  made  up  of  masses  of  iron  of  different  dimensions 
and  qualities,  the  total  magnetic  reluctance  can  be  computed 
by  summing  up  the  separate  reluctances  as  in  the  case  of  elec- 
trical resistances.  The  unit  of  reluctance  is  called  an  oersted. 

Definition. — An  oersted  is  the  reluctance  in  a  magnetic  cir- 
cuit when  one  gilbert  is  required  to  produce  a  flux  of  one  max- 
well in  the  circuit. 

Problems 

1.  What  MJM.F.  is  produced  by  a  solenoid  of  95  turns  of  wire  through 
which  flows  a  current  of  5  amperes?  596.90  gilberts. 

2.  An  iron  ring  is  uniformly  wound  with  250  turns  of  wire.     A  current 
of  4  amperes  produces  a  magnetic  flux  of  800,000  maxwells.     Find  the 
reluctance  of  the  ring.  0.00157  oersted. 

125.  Flux  from  a  Permanent  Magnet. — The  following  exer- 
cises will  illustrate  the  manner  in  which  magnetic  flux  is 
measured.  They  may  also  serve  to  make  clear  some  points 
that  perhaps  have  not  been  fully  understood  from  merely 
reading  the  theory  of  the  subject.  In  regard  to  the  second  one, 
it  is  important  that  the  facts  be  impartially  observed  and  that 
a  full  and  sufficient  reason  therefor  be  thoroughly  understood. 

A.  Place  a  coil  of  one  turn  about  the  middle  of  a  bar  magnet. 
Note  the  deflection  of  the  ballistic  galvanometer  when  the 
magnet  is  quickly  withdrawn.     Repeat,  using  two,  three,  five, 
and  ten  turns. 

B.  Does  it  make  any  difference  whether  the  coil  is  drawn  off 
the  north  or  the  south  end  of  the  magnet? 

C.  Place  the  coil  just  off  one  end  of  the  magnet  and  suddenly 
withdraw  the  latter.     Note  the  deflection.     Repeat,  moving 


156 


ELECTRICAL  MEASUREMENTS 


the  coil  instead  of  the  magnet.  Repeat  again,  moving  the 
coil  at  right  angles  to  the  magnet  so  as  to  "cut"  across  the 
magnetic  flux.  Is  it  true  to  say  that  the  deflection  depends 
only  upon  the  total  change  in  the  flux  threading  the  coil,  and 
not  at  all  upon  the  manner  in  which  that  change  is  produced? 
126.  Study  of  a  Magnetic  Circuit— Bar  and  Yoke.— The 
bar  and  yoke  apparatus  consists  of  a  heavy  rectangle  of  iron, 
through  the  center  of  which  is  the  bar  to  be  studied.  This 
bar  is  divided  in  the  middle  and  the  two  ends  surfaced  to  fit 


FIG.  71. — Bar  and  yoke. 

well  together.  One-half  of  the  bar  is  fixed  rigidly  in  place  while 
the  other  half  can  be  withdrawn  by  the  handle,  H.  Around 
the  bar  near  the  middle  is  a  small  bobbin  on  which  is  wound 
the  coil  which  is  joined  to  the  ballistic  galvanometer.  When 
the  movable  part  of  the  bar  is  withdrawn,  this  bobbin  flies 
out  to  one  side,  thus  carrying  the  coil  into  a  region  of  no,  or 
very  small,  magnetic  flux.  Thus  the  deflection  of  the  galva- 
nometer shows  the  total  flux  through  the  bar  before  it  was 
parted. 

The  magnetizing  current  is  passed  through  the  coils  FF, 
which  extend  the  entire  length  of  the  bar,  save  for  the  short 
portion  occupied  by  the  bobbin  at  the  middle.  The  bar  is 
thus  subjected  to  a  uniform  magnetizing  force  throughout  its 
entire  length.  No  magnetic  poles  are  produced  since  the  flux 


THE  MAGNETIC  CIRCUIT  157 

through  the  bar  completes  its  circuit  through  the  yoke,  the 
reluctance  of  which  is  small  in  comparison  with  that  of  the  bar. 
Thus  the  length  of  the  magnetic  circuit  for  all  practical  pur- 
poses is  equivalent  to  the  length  of  the  bar  alone. 

Starting  with  a  small  current  through  the  magnetizing  coils, 
the  corresponding  deflection  of  the  galvanometer  is  determined. 
Then  the  current  is  increased  by  small  steps  and  the  corre- 
sponding deflections  are  noted,  till  the  iron  is  fully  magnetized. 
If  the  bar  has  been  previously  magnetized  it  will  be  necessary 
to  demagnetize  it  before  using  it.  This  can  be  done  by  start- 
ing with  the  largest  current  previously  used  and  decreasing  it 
by  steps  while  at  the  same  time  it  is  slowly  reversed  in  direc- 
tion eight  or  ten  times  at  each  step. 

127.  Computation  for  Magnetomotive  Force. — The  M.M.F. 
is  readily  computed  from  the  value  of  the  current  by  the 
formula, 

M.M.F.  =  0.47TAT7. 

where  I  is  the  value  of  the  current  expressed  in  amperes. 

128.  Measurement  of  Magnetic  Flux. — The  magnetic  flux 
is  measured  by  the  deflection  of  the  galvanometer,  the  latter 
being  directly  proportional  to  the  amount  of  flux  cut  by  the 
wires  on  the  bobbin  when  it  flies  out  from  its  position  around 
the  bar.     If  <j>  denotes  the  total  flux  through  the  bar,  and  n 
the  number  of  turns  of  wire  on  the  bobbin,  the  effect  on  the 
galvanometer  is  the  same  as  cutting  a  flux  of  <t>n  with  a  single 
wire.     Therefore, 

<t>n  =  cdj 

where  d  is  the  deflection  in  scale  divisions,  and  c  is  the  magnetic 
ballistic  constant  of  the  galvanometer  expressed  in  flux  turns 
per  scale  division.  The  actual  flux  through  the  bar  is,  then, 

<f>  =  —  d. 

n 

129.  To  Find  the  Constant  c. — The  constant  c  is  readily 
determined  if  a  standard  magnet  is  at  hand.     This  is  a  steel 


158  ELECTRICAL  MEASUREMENTS 

magnet  the  total  flux  through  which  is  known.  Call  this  <£ 
maxwells.  A  coil  of  n'  turns  is  placed  around  the  middle  of 
the  magnet  and  connected  to  the  galvanometer.  When  the 
magnet  is  withdrawn  the  flux  $  is  cut  n'  times,  and  as  before, 

$  n'  =  c  d', 
from  which, 


130.  Plotting  the  Results.  —  The  results  obtained  from  this 
study  of  a  magnetic  circuit  can  best  be  shown  by  means  of 
curves.     The  first  one  may  be  plotted  taking  values  of  the 
M.M.F.  as  the  abscissae,  and  the  corresponding  values   of 
magnetic  flux  for  the  ordinates. 

From  the  values  of  the  flux  and  the  M.M.F.  compute  the 
reluctance  of  the  bar  for  different  values  of  the  flux.  Express 
the  results  by  means  of  a  second  curve,  this  time  taking  the 
flux  for  abscissas  and  using  the  corresponding  values  of  the 
reluctance  for  ordinates. 

131.  Standard  Curve  —  Unit  Circuit.  —  Inasmuch  as  the  flux 
through  a  magnetic  circuit  depends  upon  the  length  and  cross 
section  of  the  circuit,  as  well  as  upon  the  M.M.F.,  the  curve 
just  drawn  shows  the  relation  between  the  flux  and  the  M.M.F. 
for  this  particular  circuit  only.     If  it  is  desired  to  find  this 
relation  for  other  circuits  of  the  same  material  the  results 
obtained  above  must  be  expressed  in  terms  of  a  "  unit  circuit.  " 
That   is,  we   can  imagine  a  bit  of  the  circuit  1  cm.   long 
and    1    sq.   cm.    in   cross   section,    say   a   centimeter    cube, 
at    some  part   of  the   circuit.     Let   us   now    draw    a   flux- 
M.M.F.  curve  for  this  unit  circuit.      This  will  be  independent 
of  the  dimensions  of  the  whole  circuit,  and  will  represent  only 
the  characteristics  of  the  material  being  examined. 

The  ordinates  for  this  curve,  that  is,  the  flux  passing  through 
the  unit  circuit,  can  easily  be  computed  from  the  values  of  the 
total  flux  through  the  whole  circuit.  It  is  simply  the  total 
flux  divided  by  the  cross  section  of  the  circuit.  This  quotient, 


THE  MAGNETIC  CIRCUIT  159 

which  is  the  flux  per  square  centimeter,  is  usually  denoted  by 
the  letter  B,  or  in  symbols, 


where  A  denotes  the  cross  section  of  the  circuit  in  which  the 
total  flux  is  <£. 

Similarly,  the  abscissae  are  computed  from  the  values  of 
the  total  M.M.F.  Since  the  unit  circuit  is  1  cm.  long 
the  M.M.F.  in  this  length  will  be  the  total  M.M.F.  divided 
by  the  length  of  the  entire  circuit  —  provided  the  latter  is 
uniform.  This  quantity,  which  is  the  M.M.F.  per  linear  centi- 
meter, is  usually  denoted  by  the  letter  H,  or  in  symbols, 

M.M.F. 
~L~ 

where  L  is  the  length  of  the  circuit. 

In  the  case  of  the  bar  and  yoke  the  reluctance  of  the  yoke 
is  small  compared  with  that  of  the  bar,  and  therefore  the 
equivalent  length  of  the  whole  circuit  is  practically  that  of 
the  bar  alone. 

Such  a  B-H  curve  should  now  be  drawn  in  addition  to  the 
two  already  plotted  for  this  circuit. 


CHAPTER  XI 
COMPLETE  DEFINITION  OF  THE  MAXWELL 

132.  Complete  Definition  of  the  Value  of  Unit  Flux.— The 
preceding  experiment  furnishes  a  good  and  complete  study  of  a 
magnetic  circuit,  and  the  curves  just  obtained  show  the  char- 
acteristics of  the  material  composing  the  circuit.    But  referring 
back  to  the  discussion  of  magnetic  flux  it  will  be  found  that 
the  value  of  a  maxwell  of  flux  was  there  stated,  but  it  has  not 
yet  been  denned.     If  anything  more  than  an  empirical  rule 
of  thumb  is  desired,  it  will  be  necessary  to  find  a  definition  of 
unit  flux. 

Why  can  not  the  statement  in  Article  122  regarding  the 
value  of  one  maxwell  be  taken  as  a  definition,  and  thus  fix  the 
amount  of  flux  that  should  be  taken  as  unity?  Because  in  the 
definitions  for  unit  current  and  for  unit  magnetic  pole  there 
has  already  been  given,  implicitly,  the  value  of  unit  flux.  And 
the  only  consistent  thing  that  can  now  be  done  is  to  determine, 
explicitly,  just  how  much  flux  has  thus  been  fixed  at  unity. 
This  will  require  a  careful  review  of  all  the  definitions  and  other 
relations  between  magnetic  poles  and  electric  currents.  The 
discussion  is  straightforward,  except  as  we  may  pause  oc- 
casionally by  the  way  to  note  some  interesting  or  important 
facts. 

We  shall  therefore  now  take  up  in  logical  order  a  series  of 
definitions  and  propositions  leading  to  the  discovery  of  the 
value  of  unit  flux,  and  the  establishment  of  the  relations  from 
which  will  follow  the  statement  heretofore  given. 

133.  Relations  Between  Current  and  the  Magnetic  Field. — 
The  definition  for  unit  current  gives  the  relation  that  the 
amount  of  work  required  to  carry  a  unit  magnetic  pole  once 

160 


COMPLETE  DEFINITION  OF  THE  MAXWELL 


161 


around  an  electric  current  is  47r7  ergs.  For  a  pole  of  strength 
m  the  work  would  be  4fjrlm  ergs.  Since  this  is  true  for  all 
cases,  it  will  be  true  for  the  particular  case  of  a  long  straight 
current  with  the  pole  carried  around  it  in  a  circle  of  radius  a 
perpendicular  to  the  conductor,  the  center  of  the  circle  being 
at  the  center  of  the  conductor.  At  each  point  in  the  path  the 
pole  is  airways  at  the  same  distance  from  the  current,  and  what- 
ever force  it  may  experience  at  one  point  will  be,  by  symmetry, 
the  same  at  every  other  point  in  its  path.  Let  F  denote  the 
value  of  this  force.  The  length  of  the  path  is  2ira.  Therefore 
the  work  required  to  carry  the  pole  once  around  the  current 
against  this  force  is, 

W  =  2iraF. 

But  from  the  definition  of  unit  current  this  is  also, 

W    =    47T/W. 

Hence  the  force  experienced  by  the  pole  at  each  point  of  the 
path  is, 

v         2Im    j 

F  =  —^—  dynes. 

134.  Force  between  Current  and  Magnetic  Field. — When  a 

conductor  carrying  a  current  is  also  in  a  magnetic  field,  there 


(a)  (b)  (0 

FIG.  72. — The  magnetic  field  surrounding  an  electric,  current. 

is  a  force  acting  upon  the  conductor  tending  to  move  it  across 
the  field.  This  force  is  due  to  the  mutual  action  between  the 
field  produced  by  the  current  and  the  original  field  which  exists 
independently  of  the  current. 

Suppose  the  original  field  is  uniform  and  constant  when  no 
11 


162 


ELECTRICAL  MEASUREMENTS 


current  flows  through  the  straight  conductor  under  considera- 
tion: This  condition  can  be  represented  as  in  Fig.  72a.  The 
field  around  a  current  would  be  represented  by  a  series  of  con- 
centric circles  with  centers  on  the  conductor  as  shown  at  b. 
If  now  these  fields  both  exist  at  the  same  time,  the  sum  of  the 
two  is  as  shown  in  c,  and  the  conductor  is  urged  downward. 
Evidently  this  effect  is  greatest  when  the  current  is  at  right 
angles  to  the  direction  of  the  magnetic  field.  The  exact  value 
of  the  force  acting  on  the  conductor  can  be  determined  from 
the  following  considerations. 


ds 


FIG.  73. 


It  is  shown  above  that  the  force  on  a  pole,  m,  at  P,  due  to  a 
current  in  the  long  straight  conductor  AB,  is 


Since  action  and  reaction  are  equal,  the  conductor  itself 
must  experience  an  equal  and  opposite  force  tending  to  move  it 
at  right  angles  to  the  plane  APB,  and  which,  in  terms  of  the 
polar  theory  of  magnetism,  would  be  said  to  be  due  to  the  pole 


COMPLETE  DEFINITION  OF  THE  MAXWELL         163 

at  P.  But  as  there  can  be  no  "  action  at  a  distance,"  or  as 
Lord  Kelvin  used  to  put  it,  "  action  by  matter  where  it  is  not," 
it  would  be  more  consistent  to  express  this  force  in  terms  of  the 
magnetic  field  immediately  surrounding  the  conductor.  And 
although  in  reality  all  electric  currents  flow  in  closed  circuits, 
yet  for  purposes  of  computation  it  is  often  convenient  to  deal 
with  portions  of  a  circuit.  For  this  reason,  and  in  this  sense, 
the  total  force  experienced  by  the  conductor  AB  may  be  con- 
sidered as  the  sum  of  all  the  forces  on  each  part  of  the  wire. 

Thus  let  us  consider  an  element  of  length  ds  at  any  point  T. 
The  intensity  of  the  magnetic  field  at  T  due  to  a  pole  of  strength 
m  at  P  is 


and  the  component  of  this,  perpendicular  to  the  wire  A  B  is 
Hf  =  ^  cos  0  (3) 

It  is  natural  to  suppose  that  the  force  acting  upon  an  element 
of  the  conductor  at  T  will  depend  upon  the  amount  of  current 
in  the  wire,  the  intensity  of  the  magnetic  field  and  the  length 
of  the  portion  considered. 
That  is,  if  /  denotes  the  value  of  this  force, 

/  a  (I,H',ds) 
or, 

/  =  klH'ds  (4) 

where  k  is  the  proportionality  factor.  The  force  on  the 
whole  wire  will  then  be  the  summation  of  the  forces  acting 
on  each  separate  portion  of  it.  In  Eq.  (1)  we  see  what  this 
summation  must  be.  We  also  see  that  m  appears  in  this  sum- 
mation in  the  first  power;  therefore  each  of  the  other  factors 
in  H'  must  also  enter  in  the  first  power. 

The  force  acting  upon  the  entire  length  of  this  conductor 
is,  then, 

/?-l-cn 

Im  cos  6  ds 


164  ELECTRICAL  MEASUREMENTS 

The  three  variables  can  be  expressed  in  terms  of  one  by  ob- 
serving from  similar  triangles  that 

£-?  (6) 

which  gives, 

klm      +1  2klm 

F  =  -  cos  0  de  =  -  (7) 


r+1 
J  _j 


This  agrees  with  (1)  if  k  =  JJL.  Therefore  the  relation  in  (4) 
should  have  been 

/  =  nH'Ids  =  B'lds  (8) 

as  the  true  expression  for  the  value  of  the  force  acting  upon  an 
element  of  the  conductor. 

135.  Magnetic  Induction.  —  We  are  here  introduced  to  a  new 
quantity  which  now  for  the  first  time  enters  our  equations  in 
explicit  form.  This  is  the  quantity  pH,  which  often  appears 
in  this  way,  the  two  factors  entering  thus  together.  This 
product  has  distinctive  properties  which  are  quite  different 
from  those  of  H  alone,  the  same  as  the  product  RI  is  not  like  7. 
And  as  in  the  latter  case,  the  letter  V  is  often  written  for  RI, 
and  the  quantity  thus  designated  is  given  a  distinctive  name, 
"fall  of  potential,"  so  in  the  former  case  the  product  pH  is 
usually  represented  by  the  single  letter  B,  and  the  quantity 
thus  designated  is  called  the  "magnetic  induction." 

As  H  denotes  the  intensity  of  the  magnetic  field  and  has 
different  values  from  point  to  point,  so  B  has  a  definite  value 
at  each  point.  The  value  of  H  is  measured  by  the  force  which 
acts  upon  a  magnetic  pole;  the  value  of  B  is  measured,  Eq.  (8)r 
by  the  force  which  acts  upon  a  conductor  carrying  an  electric 
current.  Either  quantity  may  be  represented  by  lines  drawn 
through  each  point  in  space  in  the  direction  of  the  force  at  that 
point.  If  more  lines  are  drawn  where  the  force  is  strong,  and 
fewer  where  the  force  is  less,  the  density  of  these  lines,  that 
is,  the  number  per  square  centimeter,  may  also  be  made  to 
represent  the  numerical  value  of  H,  and  of  B,  respectively. 
This  idea  of  drawing  lines  to  represent  these  quantities  was 


COMPLETE  DEFINITION  OF  THE  MAXWELL        165 

introduced  by  Faraday,  who  used  them  thus  to  visualize  the 
state  of  the  medium  in  the  neighborhood  of  a  magnet  or  a 
current.  For  the  case  of  ju  =  1,  it  is  evident  that  the  map  for 
H  would  be  similar  to  the  map  for  B;  for  other  values  of  ju 
the  maps  for  the  two  would  be  different. 

136.  Force  Exerted  on  a  Conductor  Carrying  a  Current. — 
Equation  8,  above/ is  a  very. important  relation.     In  the  first 
place  it  shows  that  the  force  acting  on  the  conductor  does  not 
depend  solely  upon  H,  the  intensity  of  the  magnetic  field, 
but  it  is  proportional  to  pH,  which  is  quite  a  different  thing. 
Furthermore,  if  a  finite  length  L  of  the  conductor  crosses  a 
flux  of  uniform  density  B,  the  force  acting  upon  this  length  is 

F  =  B'lL,  (9) 

where  B'  (from  Eq.  3)  is  the  component  of  the  magnetic 
induction  at  right  angles  to  the  direction  of  L. 

If  in  addition  to  being  uniform,  the  magnetic  induction  is 
at  right  angles  to  the  current,  then  a  wire  L  centimeters  in 
length  will  be  urged  across  the  field  with  a  force, 

F  =  BIL  dynes.  (10) 

This  force  acts  upon  the  conductor  as  a  whole,  and  not  upon 
the  current  which  it  carries.  Several  investigations  have  been 
made  to  determine  whether  the  current  is  urged  to  one  side  of 
the  conductor,  but  no  effect  of  this  kind  has  ever  been  detected. 

Definition. — Unit  magnetic  induction  is  that  magnetic 
induction  in  which  a  conductor  carrying  unit  current  (  =  10 
amperes)  experiences  a  force  of  one  dyne  for  each  centimeter  of 
length. 

This  definition  does  not  lend  itself,  directly,  to  the  measure- 
ment of  magnetic  induction  because  it  is  not  easy  to  measure 
small  forces  with  accuracy.  But  it  does  lead  at  once  to  a  sim- 
ple method  for  measuring  the  total  magnetic  flux,  from  which 
the  magnetic  induction  is  readily  obtained,  if  desired,  by  com- 
putation, as  is  shown  below. 

137.  Work  when  a  Current  Moves  through  a  Magnetic 
Field. — Let  a  conductor  such  as  that  supposed  in  (10)  above, 


166  ELECTRICAL  MEASUREMENTS 

and  carrying  a  current  7,  move  a  distance  s  in  the  direction  in 
which  it  is  urged.  The  work  which  can  be  done  by  F  is 

W  =  Fs  =  BLIs,  (1) 

and  this  energy  is  supplied  by  the  source  that  is  maintaining 
the  current. 

As  explained  above,  B  denotes  the  value  of  the  magnetic 
induction  at  a  given  point.  When  multiplied  by  a  definite 
area,  normal  to  the  direction  of  B,  the  product  is  the  total 
induction,  or  the  magnetic  flux,  through  this  area.  Since 
Ls  is  the  area  swept  over  by  the  conductor,  the  product  BLs 
is  the  magnetic  flux,  $,  cut  by  the  conductor  in  its  motion. 
Therefore, 

W  =  <t>I  (2) 

from  which  it  appears  that  one  erg  of  work  is  required  to 
move  a  conductor  carrying  unit  current  across  a  magnetic  flux 
of  one  maxwell. 

138.  Induced  Electromotive  Force. — The  electrical  express- 
ion for  work  is 

Work  =  Elt  =  («'  +  e)  It 
=  RPt  +  elt, 

where  E  is  the  E.M.F.  required  to  maintain  the  current  7,  and 
t  is  the  time  that  the  current  is  flowing,  measured  in  seconds. 
A  part  the  energy  is  used  in  merely  keeping  the  current  flowing 
through  the  resistance  in  the  circuit  in  accordance  with  Ohm's 
law;  and  this  part  appears  as  heat.  If  any  other  work  is 
done  it  is  included  in  the  part 

W  =  elt, 

where  e  is  the  extra  E.M.F.  required  to  do  the  work.  Since  e  is 
not  constant  it  is  better  to  write, 

W  =   (eldt  =  07.  (3) 

Equating  (1)  and  (3), 

BLs  =  0.  (4) 


/•*.- 


COMPLETE  DEFINITION  OF  THE  MAXWELL         167 
Differentiating  this  gives  for  the  value  of  e, 


Therefore  the  movement  of  the  conductor  across  the  mag- 
netic flux  has  induced  in  the  conductor  itself  an  equal  and 
opposite  E.M.F.,  viz., 


and  it  is  to  counter  balance  this  induced  E.M.F.  that  it  is 
necessary  to  supply  the  extra  E.M.F.  to  keep  the  current  from 
decreasing. 

Since  Eq.  (6)  is  independent  of  the  value  of  the  current  in 
the  wire,  e  will  have  the  same  value  when  the  current  is  zero. 
Therefore  when  a  wire  carrying  no  current  is  moved  across  a 
magnetic  flux  there  is  induced  in  it  an  E.M.F.  the  same  as 
above,  and  if  the  circuit  is  closed  this  will  cause  a  current  to 
flow.  The  direction  of  this  current  will  be  opposite  to  that 
of  the  steady  current  considered  in  (3). 

139.  Definition  of  a  Maxwell.  —  As  stated  in  Article  122,  the 
unit  of  magnetic  flux  is  called  a  maxwell.  Equation  (6)  now 
shows  that  a  magnetic  flux  can  be  measured  by  means  of  the 
E.M.F.  induced  in  a  conductor,  and  this  relation  furnishes  the 
basis  for  defining  the  value  of  a  maxwell. 

Definition.  —  A  maxwell  is  the  magnetic  flux  cut  each  second 
by  a  conductor  in  which  there  is  induced  an  E.M.F.  of  one 
C.G.S.  unit. 

The  name  "  maxwell  "  for  unit  magnetic  flux  was  adopted 
by  the  International  Electrical  Congress  at  Paris  in  1900.  l 
Where  108  maxwells  are  cut  each  second  the  induced  E.M.F.  is 
one  volt. 

Corollary.  —  Unit  Flux  Density.  A  uniform  flux  of  one 
maxwell  through  each  square  centimeter  normal  to  the  direc- 
tion of  the  flux  would  be  a  flux  density,  or  magnetic  induction, 
of  unity. 

Equation  10,  Article  136,  also  gives  the  value  of  this  unit 

]  See  Elec.  Rev.,  Vol.  47,  p.  441.    1900. 


168  ELECTRICAL  MEASUREMENTS 

expressed  in  terms  of  the  force  exerted  upon  a  conductor  carry- 
ing a  current. 

140.  Measurement  of  Magnetic  Flux. — Having  seen  above 
that  an  E.M.F.  is  induced  in  a  wire  when  the  latter  cuts  across 
a  magnetic  flux,  it  is  at  once  evident  that  this  offers  a  ready 
means  for  the  measurement  of  magnetic  flux.  As  this  E.M.F. 
exists  only  while  the  flux  is  being  cut,  it  gives  rise  to  a  transient 
current,  the  total  quantity  in  which  can  be  measured  by  a 
ballistic  galvanometer,  as  shown  below. 

The  equation  of  a  ballistic  galvanometer  is, 

Q  =  kd, 

where  ft  is  a  constant  and  d  is  the  first  throw  of  the  needle 
corrected  for  damping.  If  the  galvanometer  is  connected  to 
the  wire  when  it  is  moved  across  a  magnetic  flux,  the  total 
quantity  of  electricity  that  passes  through  the  circuit  is, 

(7) 


R 

where  <£"  denotes  the  amount  of  flux  enclosed  by  the  galvanome- 
ter circuit  at  the  beginning,  and  <£'  the  amount  at  the  end  of 
the  motion.  Hence  the  amount  of  flux  cut  across  by  the  wire 
is, 

<t>'  -  <f>"  =  Rkd  (8) 

where  R  is  the  total  resistance  of  the  galvanometer  circuit. 

Hence  the  quantity  measured  by  the  ballistic  galvanometer 
in  Article  128  is  what  has  been  denoted  above  by  <£,  and  called 
magnetic  flux.  (=  ^HA.) 

141.  Relation  Between  Field  Intensity  and  M.M.F.  — Since 
a  magnetic  pole  in  a  magnetic  field  is  under  the  action  of  a 
force,  whenever  such  a  pole  is  moved  work  must  be  done,  either 
positive  or  negative  according  to  the  direction  of  the  motion. 
The  longer  the  path  and  the  stronger  the  field  the  greater  will 
be  the  expenditure  of  work. 

By  definition,  the  magnetomotive  force  in  any  circuit  is 
measured  by  the  work  per  unit  pole  to  carry  a  north  pole  once 


COMPLETE  DEFINITION  OF  THE  MAXWELL         169 

along  the  circuit.  If  the  magnetic  intensity  has  the  value  H 
at  each  point  along  the  path,  it  is  evident  that  the  work  per 
unit  pole  and  therefore  the  magnetomotive  force,  is, 


J 


HdL  =  M.M.F. 


where  L  is  the  length  of  the  path.  In  case  H  has  the  same 
value  at  each  point  this  "line  integral"  becomes  simply  HL. 
Therefore  if  enough  is  known  about  the  circuit  to  determine 
the  values  of  //,  the  M.M.F.  can  be  computed.  Or,  if  the 
M.M.F.  is  known  it  is  possible  to  compute  the  value  of  H  in 
some  simple  circuits.  A  few  such  examples  are  given  below, 
including  those  of  most  importance  in  practical  measurements. 

142.  Long  Straight  Current.  —  In  the  case  of  a  long  straight 
conductor  carrying  a  current  it  is  not  difficult  to  compute  the 
intensity  of  the  magnetic  field  at  a  distance  a  from  the  conduc- 
tor.    Since  the  work  per  unit  pole  required  to  carry  a  magnetic 
pole  around  the  current  by  any  path  whatsoever  is  4wl,  let 
the  path  be  a  circle  of  radius  a.     By  symmetry,  the  magnetic 
force  is  constant  along  this  path,  and  therefore 

4T/  =  H  X  2ira  (1) 

where  H  denotes  the  intensity  of  the  magnetic  field. 
From  this, 

-! 

143.  Ring    Solenoid.  —  Another    case    that    can    be    easily 
solved  is  that  of  a  uniform  spiral,  or  helix,  of  many  turns  of 
wire  and  bent  into  a  circle  so  as  to  bring  its  ends  together. 
This  is  the  case  of  a  ring  uniformly  wound  with  N  turns  of 
wire.     The  magnetic  field  is  all  within  the  spiral  forming  the 
ring  and  in  the  direction  of  its  length.     It  will  therefore  require 
work  to  carry  a  magnetic  pole  around  the  ring,  for  in  making 
the  complete  circuit  the  pole  has  passed  around  a  total  cur- 
rent of  NI.    The  total  work  per  unit  pole  is,  then, 


m 


170  ELECTRICAL  MEASUREMENTS 

If  the  path  is  a  symmetrical  circle  of  radius  r,  the  magnetic 
forces  will  have  the  same  value,  H,  at  each  point  of  the  path. 
Hence  if  L  is  the  length  of  the  path, 

W  =  HmL, 
and 

H  =  47rv-/  =  47rtt/, 

where  n  denotes  the  number  of  turns  per  centimeter. 
Since  L  =  2irr,  where  r  is  the  radius  of  the  path, 

4--7V7       A/-/' 

TT  T/l  J.  V  J.  J.1  JL 

H  =  -2*r=-fr' 

where  /'  is  the  value  of  the  current  in  amperes.  This  shows 
that  the  intensity  of  the  magnetic  field  is  not  constant  across 
the  section  of  the  ring,  but  varies  inversely  as  r,  being  greater 
on  the  inner  side  of  the  ring. 

144.  Long  Straight  Solenoid. — A  long  straight  solenoid, 
wound  uniformly  with  n  turns  per  centimeter,  may  be  con- 
sidered as  a  portion  of  a  ring  solenoid  of  very  great  radius.  At 
points  within  the  solenoid,  and  not  near  the  ends,  the  magnetic 
forces  are  practically  the  same  as  though  the  entire  ring  were 
present.  From  the  preceding  section  the  value  of  the  magnetic 
intensity  is,  then, 

H    =   47TTC/. 

Or,  if  the  current  is  measured  in  amperes 
H  =  1.2566n/'. 

146.  Magnetic  Field  due  to  Any  Electric  Current. — In  the 

examples  considered  above  the  magnetic  field  due  to  a  current 
could  be  very  easily  calculated.  In  most  cases  the  calculation 
is  not  so  simple.  But  if  we  may  make  the  assumption  that  each 
linear  element  of  current,  independently  of  the  other  elements 
of  the  circuit,  contributes  its  own  share  toward  making  up 
the  total  field,  it  is  possible  to  find  an  expression  for  the  field 
due  to  a  single  element  which,  when  integrated  over  the  entire 


COMPLETE  DEFINITION  OF  THE  MAXWELL         171 

length  of  the  electric  circuit,  will  give  the  correct  value  for  the 
intensity  of  the  magnetic  field  at  any  given  point.  This  will 
enable  us  to  calculate  the  magnetic  field  due  to  a  current  in  any 
circuit  whatever.  The  results  so  obtained  have  been  found  to 
agree  with  the  facts  in  every  case  where  the  magnetic  field  can 
be  measured  by  other  means.  This  is  not  to  be  taken  as 
proving  what  the  effect  of  a  single  element  would  be,  if  such  an 
element  could  exist  by  itself,  but  as  showing  that  the  formula 
will  give  correct  results  when  applied  to  any  actual  current. 

146.  Magnetic  Effect  of  a  Current  Element.— Let  ds, 
Fig.  74,  represents  a  short  element  of  an  electric  circuit.  It  is 
required  to  find  the  intensity  of  the  magnetic  field  at  any  point, 
P,  due  to  a  current  I  flowing  through  ds. 

By  the  definition  of  Article  3  the  intensity  of  the  field  at  P 
is  measured  by  the  force  per  unit  pole  which  would  be  exerted 
upon  a  magnetic  pole  if  one  were  placed  at  P.  That  is, 

«-£.  (i) 

where  dH  denotes  the  part  of  the  total  intensity  that  is  due 
to  the  element  ds.  Therefore  let  us  suppose  there  is  a  pole  m 
at  P,  and  then  let  us  determine  the  force  which  it  would 
experience.  Finding  F,  the  value  of  H  is  readily  computed. 
In  Eq.  8,  Article  134,  above,  the  force  which  a  linear  current 
element  would  experience  in  a  magnetic  field,  H',  at  right 
angles  to  itself,  was  found  to  be 

/  =  »H'Ids  (3) 

In  order  to  express  this  force  in  terms  of  the  pole  m,  the 
value  of  H'  as  found  on  page  163  may  be  substituted  in  (3). 
This  gives 

-  __  mlds  cos  0.  ,£. 

J  ~          r2  v*' 

Since  action  and  reaction  are  equal,  if  the  element  ds  ex- 
periences this  force  because  of  the  presence  of  m,  then  m  will 


172 


ELECTRICAL  MEASUREMENTS 


experience  an  equal  force  of  reaction  from  ds.    Therefore 
the  force  acting  upon  m  is, 


F  = 


H' 


and  the  part  of  the  magnetic  field  at  P  contributed  by  the 
element  ds  is, 

,TT      F       I  ds  cos  0  //A 

aH  =  —  = (6; 

m  r2 

The  expression  ds  cos  0  is  readily  seen  to  be  the  component 
of  ds  normal  to  r. 

147.  Extension  to  General  Case. — To  find  the  value  of  H 

at  any  given  point  P,  due  to  a  current 
in   a   given   circuit,  let  P,  Fig.  74,  be 
[\  -  -  \  taken  as  the  origin  of  polar  coordinates, 

r  and  0.  Then  the  integral  of  (6)  taken 
along  the  entire  length  of  the  electric  cir- 
cuit will  give  the  correct  value  for  the 
field  intensity  at  P.  The  computation 
will  be  simpler,  in  general,  if  the  entire 
circuit  and  P  are  in  one  plane,  for  then 
all  of  the  components,  such  as  dH,  will 
be  in  the  same  straight  line  through  P 
and  normal  to  the  plane,  and  the  re- 
sultant will  be  merely  their  sum. 

If  the  various  elements,  dH,  do  not  lie 
in  the  same  line,  the  resultant  field  in 

any  direction  is  the  sum  of  the  components  of  the  various  dH's 

resolved  in  the  given  direction. 

148.  Magnetic  Field  at  Center  of  a  Circle. — In  the  case  of  a 
circular  loop  of  current  with  the  point  P  taken  at  the  center  of 
the  circle,  the  above  integration  can  be  performed  quite  simply. 
Each  element,  ds,  of  the  conductor  is  now  at  the  same  distance, 
r,  from  P,  and  since  ds  is  always  at  right  angles  to  r,  cos  0  is 
unity  for  each  element.     Since  P  lies  in  the  plane  of  the  circle, 


FIG.  74. 


COMPLETE  DEFINITION  OF  THE  MAXWELL          173 

the  resultant  magnetic  force  is  simply  the  arithmetical  sum 
of  the  effects  from  each  element  of  the  current.     Therefore, 


for  the  entire  circle. 

COROLLARY. — The  common  definition  of  unit  current  fol- 
lows at  once  from  this  as  that  current  which  will  produce  at 
the  center  of  a  circle  of  unit  radius,  a  magnetic  field  of  unit 
intensity  for  each  centimeter  length  of  the  current. 

Problem. — Given  a  current  of  /  amperes  flowing  in  a  circle  of  radius 
a  cms.  Find  the  value  of  H  at  a  point  on  the  axis,  2a  cms.  from  the 
plane  of  the  circle. 


CHAPTER  XII 

MAGNETIC  TESTS  OF  IRON  AND  STEEL 

149.  Introduction. — In   the   preceding   chapter   there   was 
given  one  method  for  determining  the  magnetic  qualities  of  a 
piece  of  iron.     The  bar  and  yoke  method  is  very  useful  for 
purposes  of  illustration  and  instruction,  since  it  is  readily 
understood  that  the  wire  on  the  little  bobbin  cuts  across  the 
magnetic  flux  when  it  is  released  by  the  withdrawal  of  the  bar. 
But  the  air  gaps  at  the  end  of  the  bar  and  at  the  side  where  it 
slides  in  the  yoke  introduce  considerable  reluctance  into  the 
magnetic  circuit,  and  therefore  the  magnetic  flux  is  smaller 
than  it  would  be  for  a  similar  circuit  containing  no  air  gaps. 
Such  closed  circuits  are  used  in  the  methods  described  below, 
and  while  it  is  not  possible  for  the  wire  to  cut  across  the  flux, 
yet  when  the  flux  is  changed  from  one  direction  to  the  reverse 
it  is  evident  that  the  change  in  the  flux  through  the  coil  has 
been  twice  the  original  amount.     In  fact,  if  the  bobbin  in  the 
bar  and  yoke  had  remained  around  the  bar  while  the  magnet- 
izing current  is  reversed,  the  deflection  of  the  galvanometer 
would  have  been  twice  the  amount  that  it  was  when  the  bar  was 
withdrawn. 

The  following  methods  are  of  varying  intrinsic  value,  but 
they  are  described  in  the  given  order  as  each  one  leads  up  to 
the  one  that  follows.  There  is  sufficient  repetition  to  empha- 
size the  important  points  and  to  make  each  method  intelligible 
when  that  section  only  is  read.  The  methods  given  in  Articles 
150,  155,  and  158,  are  most  frequently  used. 

150.  Double  Bar  and  Yoke. — This  apparatus  consists  of  two 
massive  yokes  of  Swedish  iron  fitted  to  carry  two  round  bars 
of  the  iron  or  steel  to  be  tested.     When  clamped  together  the 

174 


MAGNETIC  TESTS  OF  IRON  AND  STEEL 


175 


bars  and  yokes  form  a  rectangular  circuit,  the  bars  forming  the 
longer  sides.  Over  the  entire  length  of  each  bar  is  a  brass 
spool  on  which  is  wound  300  turns  of  wire  to  carry  the  current 
used  to  magnetize  the  iron.  The  joints  where  the  bars  pass 
through  the  yokes  are  carefully  fitted  to  avoid  any  unnecessary 
reluctance  at  these  places.  Thus  the  magnetic  circuit  consists 
of  the  two  bars  and  very  little  else  since  the  reluctance  of  the 
yokes  is  small  in  comparison. 

This  method  differs  from  Hopkinson's  bar  and  yoke  in  that 
the  bar  can  not  be  pulled  open  to  allow  the  test  coil  to  fly  out 
and  cut  the  flux  it  is  desired  to  measure.  Therefore  the  test 


E 


(a) 

End  View 


Bar 

o 

O 

Bar 

(b)    Top  View 

FIG.  75. — Double  bar  and  yoke,  without  the  magnetizing  coils. 


coil  is  wound  around  the  middle  of  the  bars,  and  the  flux 
through  the  circuit  is  reversed  by  reversing  the  magnetizing 
current.  This  gives  twice  the  change  of  flux  through  the  coil 
and  therefore  twice  the  deflection  that  would  be  obtained  by 
simply  cutting  the  flux  once. 

In  making  a  test  by  this  method  it  is  well  to  start  with  a 
magnetizing  .current  of  about  one  ampere.  This  should  be 
reversed  ten  or  more  times,  allowing  the  current  to  rise  to  its 
full  value  after  each  reversal.  On  the  last  reversal  the  galvan- 
ometer circuit  can  be  closed  and  the  deflection  observed.  This 
should  be  repeated  twice,  or  until  consistent  readings  are  ob- 
tained, closing  the  galvanometer  circuit  only  when  the  reversal 
is  the  same  way  as  before  in  order  to  keep  the  galvanometer 
deflection  always  in  the  same  direction.  The  current  can  now 
be  reduced  about  10  per  cent,  and  after  eight  or  ten  slow 


176  ELECTRICAL  MEASUREMENTS 

reversals,  as  before,  the  corresponding  deflection  can  be 
observed.  In  this  way  the  current  is  reduced  to  zero  by  ten  or 
fifteen  steps,  which  should  be  shorter  when  going  over  the  steep 
portion  of  the  curve.  If  when  adjusting  the  current  to  a  new 
value  it  falls  too  low  it  can  be  raised  to  the  desired  value  and 
no  harm  is  done.  On  the  other  hand,  should  the  current, 
even  for  an  instant,  reach  a  larger  value  than  the  one  last  used 
it  will  be  necessary  to  reduce  it  again  by  small  steps  and  many 
reversals  in  the  same  manner  as  was  done  the  first  time. 

The  results  of  this  test  are  best  shown  by  a  magnetization 
curve.  This  could  be  plotted  between  M.M.F.  for  abscissae 
and  flux  for  ordinates  as  was  done  in  a  previous  experiment. 
Or  better  still,  and  as  is  more  commonly  done,  by  plotting  the 
B-H  curve.  This  curve  has  the  same  form  as  the  other,  but 
since  the  dimensions  of  the  bar  are  divided  out,  the  B-H 
curve  shows  the  quality  of  the  iron  while  the  former  curve 
represented  a  particular  curcuit  composed  of  this  kind  of 
iron. 

151.  To  Determine  the  Constant  of  the  Galvanometer. — 
The  magnetic  ballistic  constant  of  the  galvanometer  and  the 
test  coil  can  be  determined  by  means  of  a  standard  magnet  as 
explained  in  Article  129,  or  by  using  a  "  standard  coil."  This 
is  a  mutual  inductance  consisting  of  two  coils  wound  upon  the 
same  spool.  One  of  these  coils  is  placed  in  the  galvanometer 
circuit  and  the  other  may  be  connected  to  a  battery  when  it 
is  to  be  used. 

Let  I'  denote  the  current  through  the  primary  when  the 
circuit  is  closed.  Then  the  flux-turns  through  the  secondary 
is  MI'  (see  Article  171),  where  M  is  the  mutual  inductance 
(C.  G.  S.)  of  the  given  pair  of  coils,  and  is  numerically  equal  to 
the  flux  threading  the  secondary  when  unit  (C.  G.  S.)  current 
flows  in  the  primary.  Each  time  the  primary  current,  /', 
is  made  or  broken,  the  secondary  is  cut  by  this  flux,  causing  a 
fling  dr  of  the  galvanometer. 

If  c  denotes  the  magnetic  ballistic  constant,  that  is,  the  flux 
cut  per  scale  division  of  deflection,  we  can  write, 


MAGNETIC  TESTS  OF  IRON  AND  STEEL  177 

MP 


=  MP  or, 


d' 


152.  To  Find  the  Value  of  B  from  the  Deflection.  —  In  study- 
ing a  given  sample  of  iron  or  steel  we  are  usually  not  so  much 
interested  in  the  properties  of  the  particular  piece  under 
investigation  as  in  the  specific  properties  of  that  grade  of  iron. 
The  actual  total  magnetic  flux,  0,  depends  as  much  upon  the 
dimensions  of  the  sample  studied  as  upon  the  quality  of  the 
iron.  If  the  cross  section  of  the  iron  is  A  sq.  cm. 
and  the  flux  density  is  B,  then  0  =  BA  maxwells.  It  is  this 
quantity  B  which  depends  upon  the  quality  of  the  iron,  and  in 
which  therefore  we  are  most  interested.  Since  <£  is  measured 
in  maxwells,  B  is  given  in  maxwells  per  square  centimeter, 
and  is  called  "  the  magnetic  induction."  It  is  also  called  "  flux 
density." 

From  what  has  been  said  thus  far  it  will  be  seen  that  the 
magnetic  flux  can  not  be  measured  directly.  It  is  only  the 
change  in  influx  turns  that  affects  the  galvanometer;  and  the 
value  of  the  total  flux  must  be  inferred  from  such  measure- 
ments. The  most  usual  change  of  flux  is  that  produced  by 
reversing  the  magnetizing  force.  It  is  then  assumed  that  the 
flux  is  also  reversed,  and  therefore  the  change  produced  is 
twice  the  total  flux,  Then, 

Change  in  flux  turns  =  2$n  =  2BAn  =  cd, 
or 

7?  - 

~ 


All  of  these  factors  can  be  found  and  the  numerical  value  of  J 
computed  once  for  all.  If  it  is  possible  to  adjust  the  resistance 
of  the  galvanometer  circuit  it  simplifies  the  computation  to 
make  /  =  100.  The  values  of  B  can  then  be  obtained  as  fast 
as  the  deflections  can  be  read  and  multiplied  by  J. 

153.  To  Determine  the  Value  of  H.—  The  M.M.F.  required 

to  magnetize  a  bar  of  iron  to  a  certain  degree  depends  directly 
12 


178  ELECTRICAL  MEASUREMENTS 

upon  the  length  of  the  bar.  Therefore  if  we  are  studying  the 
properties  of  the  material  rather  than  those  of  the  bar  as  a 
whole,  the  applied  magnetizing  force  is  expressed  in  terms  of 
the  M.M.F.  per  centimeter  length.  Thus, 

Total  M.M.F.         1.26  NT 
H  " "   Total  length  "IT"  gllberts  per  centimeter. 

164.  Permeability. — As  in  the  case  of  electric  conductors  the 
conductivity  of  a  substance  is, 

~  _    I/A    _  Current  density 
:  E/L  ~       Volts  per  cm. 

so  in  magnetism  the  permeability  of  a  substance  is  given  by 

the  similar  relation, 

,.  d>/A          B  Flux  density 

Permeability  =  M  =  MMfjL-  g  =  Gilberts  per  cm. 

Permeability  curves  should  also  be  drawn  for  each  kind  of  iron, 
using  the  "magnetic  induction,"  B,  or  flux  density,  for 
abscissae  and  the  corresponding  values  of  the  permeability  for 
ordinates. 

156.  The  Ring  Method. — In  some  respects  this  method  is 
preferable  to  the  bar  and  yoke.  The  sample  of  iron  under 
investigation  is  in  the  form  of  a  ring,  and  has  no  ends  and  no 
joints.  This  ring  is  wound  uniformly  along  the  entire  length 
with  one  or  more  layers  of  wire  for  the  primary  or  magnetizing 
coil.  Over  this  is  wound  the  secondary,  usually  consisting 
of  a  few  turns,  which  may  be  either  well  distributed  or  bunched 
at  one  place.  Since  the  primary  turns  are  closer  together  on 
the  inside  of  the  ring  than  on  the  outside,  the  magnetizing 
force  will  not  be  uniform  within  the  iron,  but  will  be  stronger 
near  the  inner  side.  For  this  reason  it  is  best  to  use  a  broad 
and  flat  ring,  shaped  like  a  wagon  tire.  In  selecting  the  iron, 
care  must  be  used  to  obtain  a  homogeneous  bar.  Rowland 
suggests  that  it  is  better  to  have  it  welded  than  forged  solid; 
it  should  then  be  well  annealed  and  afterward  have  the  outside 
taken  off  oil  round  to  about  one-eighth  of  one  inch  deep  in  a 


MAGNETIC  TESTS  OF  IRON  AND  STEEL  179 

lathe.  This  is  necessary,  because  the  iron  is  "burnt"  to  a  con- 
siderable depth  by  heating  even  for  a  moment  to  a  red  heat, 
and  the  permeability  of  this  portion  is  therefore  unlike  the  rest 
of  the  ring. 

If  the  ring  is  a  new  one  which  has  never  been  magnetized, 
the  smallest  currents  should  be  first  used.  Great  care  must 
be  observed  that  at  no  time  does  the  current  exceed,  even  for 
an  instant,  the  value  being  used  at  the  time.  If  the  iron  has 
been  previously  magnetized  it  must  be  thoroughly  demagnet- 
ized before  it  is  used. 

From  the  magnetizing  current  and  the  length  of  the  ring,  the 
value  of  the  average  magnetic  force  within  the  iron  is  easily 
computed  from  the  expression 

47TJV7    _NI 
''  10  L*    =  5a 

where  a  is  the  mean  radius  of  the  ring. 

From  the  galvanometer  deflection  and  the  cross  section  of 
the  ring  the  magnetic  induction  is 

cd 

"2~n~A 

as  shown  in  Article  152. 

The  results  of  this  experiment  are  best  shown  by  means  cf  a 
curve,  plotting  for  abscissae  the  values  of  H,  and  for  ordinates 
the  corresponding  values  of  B. 

156.  The  Step  by  Step  Method. — This  method  is  much  like 
the  preceding,  the  only  difference  being  that  the  magnetizing 
current  is  not  reversed  to  give  the  galvanometer  reading. 
Instead  of  this  the  current  is  constantly  maintained  in  one 
direction.  When  the  current  is  first  turned  on,  the  iron  is 
magnetized  to  the  corresponding  amount  and  if  the  galvan- 
ometer circuit  is  closed  there  will  be  a  proportional  deflection. 
When  the  galvanometer  has  returned  to  its  zeto  position  the 
current  is  suddenly  increased  to  its  next  value.  The  corre- 
sponding deflection  of  the  galvanometer  measures  not  the  actual 
magnetism  of  the  ring,  but  the  increase  over  the  former 


180  ELECTRICAL  MEASUREMENTS 

amount.  In  the  same  way  the  current  is  increased  by  steps 
until  its  maximum  value  is  reached,  while  each  of  the  corre- 
sponding deflections  are  carefully  noted.  The  actual  magnetism 
of  the  ring  at  any  stage  is  measured  by  the  sum  of  all  the  de- 
flections up  to  that  point.  If  this  sum  is  denoted  by  Sd,  the 
magnetic  induction  is 

B  =  — - 

Since  the  ammeter  in  the  primary  circuit  gives  the  total 
current  at  any  point,  the  expression  for  the  magnetic  force  will 
be  the  same  as  before. 

H 


10L 

The  H-B  curve  plotted  from  these  values  should  be  the  same, 
very  closely,  as  that  obtained  by  the  ring  method  using  re- 
versals of  the  current. 

157.  Hysteresis — Step  by  Step  Method. — In  the  step  by  step 
method  just  described  great  care  was  observed  never  to  have  a 
larger  current  in  the  primary  coil  than  that  being  used  at  the 
time,  and  the  magnetization  curve  was  obtained  by  always 
using  increasing  values  of  the  current. 

Suppose  that  after  reaching  the  maximum  the  current  should 
be  decreased  by  steps,  and  the  corresponding  deflections  noted. 
Would  the  magnetization  curve  be  retraced,  or  would  a  new 
curve  be  obtained?  As  the  current  is  slowly  removed  it 
would  be  found  that  the  magnetization  of  the  iron  does  not 
decrease  to  its  former  values,  and  when  the  current  is  reduced 
to  zero  there  still  remains  a  large  amount  of  " residual"  mag- 
netism. This  return  curve  can  be  traced  perfectly  well  by  the 
step  by  step  method,  and  it  is  shown  by  the  curve  ,AD  Fig.  76. 

It  will  even  require  the  application  of  a  reversed  magnetiz- 
ing force,  equal  to  CG,  to  reduce  the  magnetization  to  zero. 
This  value  of  H  is  called  the  coercive  force  of  the  iron.  It  is 
large  for  hard  iron,  and  steel,  but  small  for  soft  iron  and 
silicon-iron  alloys.  If  the  reversed  field  is  increased  to  a  value 


MAGNETIC  TESTS  OF  IRON  AND  STEEL 


181 


CF,  equal  to  CE,  the  iron  will  be  magnetized  as  strongly  as 
before,  but  in  the  opposite  direction,  and  it  will  hold  this 
magnetization  just  as  persistently  as  the  other.  If  H  is 
reduced  to  zero  and  again  increased  to  CE,  the  magnetization 
follows  as  shown  by  the  curve  PJA.  This  lagging  of  the 
values  of  B  behind  the  corresponding  changes  in  H  is  called 
hysteresis,  from  a  Greek  word  meaning  "to  lag  behind." 
The  complete  curve  as  thus  drawn  between  B  and  H  is  called 
a  hysteresis  curve. 


f  d       O  b  L 

FIG.  76. — Hysteresis  curve  for  tool  steel. 

For  the  determination  of  a  hysteresis  curve  by  this  method 
the  setup  and  manipulation  would  be  the  same  as  in  the  pre- 
ceding experiment.  And  the  results,  plotted  on  a  B-H  dia- 
gram will  give  a  curve  similar  to  the  one  shown  in  Fig.  76. 

158.  Hysteresis  by  Direct  Deflection. — Referring  to  Fig.  76, 
suppose  that  the  iron  has  been  carried  around  the  cycle  ADPJA 
several  times  and  left  in  the  condition  represented  by  the  point 
a.  If  then  it  is  carried  from  a  to  P  by  a  single  step,  the  deflec- 
tion of  the  galvanometer  will  measure  the  corresponding  change 
in  the  magnetic  induction,  represented  by  the  ordinate  ab. 
By  carrying  the  iron  around  the  cycle  to  the  point  a  again, 


182 


ELECTRICAL  MEASUREMENTS 


this  reading  can  be  repeated  as  many  times  as  desired,  or,  by 
stopping  at  some  other  point  near  a,  the  corresponding  ordi- 
nate  can  be  determined.  Thus  the  curve  ADP  can  be  traced. 
Let  the  setup  be  made  as  shown  in  Fig.  77,  where  Z  is  the 
ring  of  iron  to  be  studied.  This  iron  is  magnetized  by  a  current 
from  a  few  cells  of  the  storage  battery.  The  resistance  R  is  set 
at  the  value  which  gives  the  maximum  current  that  is  desired. 
S  is  an  ordinary  double  throw,  two  pole,  switch  which  is  made 
into  a  reversing  switch  by  the  addition  of  two  diagonal  con- 


^> — MAH 


FIG.  77. — To  determine  the  hysteresis  curve  for  the  ring,  Z. 

nections.  As  shown  in  the  figure,  one  of  these  diagonal  con- 
nections of  S  is  formed  by  the  adjustable  rheostat  r,  which  can 
be  short  circuited  by  closing  K.  The  rheostat  R  can  be  con- 
nected to  either  end  of  the  rheostat  r  by  closing  the  switch  T  on 
either  point  a  or  point  c.  This  introduces  r  into  the  battery 
circuit  when  S  is  up,  or  down,  respectively. 

With  T  closed  on  a. — When  the  switch  S  is  thrown  down,  thus 
connecting  the  primary  of  Z  directly  to  the  battery  circuit,  the 
magnetic  state  of  the  iron  is  represented  by  the  point  P,  Fig. 
76.  When  it  is  thrown  up,  the  current  through  Z  is  reversed, 


MAGNETIC  TESTS  OF  IRON  AND  STEEL  183 

but  the  current  will  be  smaller  than  before  because  the  extra 
resistance  r  is  now  in  the  circuit,  and  the  iron  will  be  brought  to 
some  point  as  /  on  the  curve  JA .  By  closing  K  the  iron  is 
brought  to  A,  and  when  K  is  opened  the  iron  comes  to  some 
point  as  a,  depending  upon  the  setting  of  r.  The  resistance  in 
r  must  have  been  previously  adjusted  to  the  proper  value  in 
order  that  the  iron  may  be  brought  to  the  desired  point  when  K 
is  opened.  (In  case  r  is  adjusted  after  K  is  opened,  the  iron 
must  be  taken  around  the  entire  cycle  several  times  before  it  is 
certain  that  the  point  a  is  reached.) 

The  galvanometer  is  joined  in  series  with  the  secondary  coil 
of  the  standard  mutual  inductance,  M,  a  resistance  box  .Q, 
and  the  few  turns  of  wire  forming  the  secondary  winding  on 
the  ring.  It  is  often  convenient  to  use  a  shunt  on  the  gal- 
vanometer, adjusted  to  give  critical  damping.  If  S  is  now 
thrown  down,  changing  the  value  of  H  from  +Ce  to  —CF,  the 
galvanometer  deflection  will  measure  the  corresponding  change 
in  B,  indicated  by  ab,  Fig.  76.  In  order  to  get  back  to  the  top 
of  the  curve  for  another  determination,  S  is  thrown  up  bringing 
the  iron  to/  on  the  lower  curve.  Then  K  is  closed,  thus  carry- 
ing the  iron  on  to  A .  Opening  K  brings  the  iron  to  the  point 
on  AD  corresponding  to  the  setting  of  r. 

The  cycle  of  operations  is  thus,  (1)  throw  S  down,  (2)  throw 
S  up,  (3)  close  K,  (4)  open  K.  The  galvanometer  is  read  dur- 
ing the  first  operation. 

With  T  closed  on  c. — To  locate  points  on  the  curve  between 
D  and  P  requires  a  slight  change  in  the  relative  position  of  r. 
This  is  effected  by  changing  the  connection  at  T  from  a  to  c. 
Then  when  S  is  thrown  up  the  iron  is  always  at  A,  Fig.  76, 
and  when  thrown  down  the  current  will  be  reversed  but  less 
than  its  maximum  value  because  r  is  now  in  the  circuit.  By 
setting  r  at  the  proper  value  the  iron  can  be  brought  to  any 
desired  point  between  D  and  P.  When  K  is  closed  the  iron  is 
carried  on  to  P  and  the  corresponding  deflection  of  the  gal- 
vanometer gives  the  ordinate  cd. 

The  cycle  of  operations  is  now,  (1)  throw  S  up,  (2)  throw  S 


184  ELECTRICAL  MEASUREMENT 

down,  (3)  close  K,  (4)  open  K.     The  galvanometer  is  read  at 
the  time  of  the  third  operation. 

In  all  of  these  measurements  the  iron  is  carried  to  P,  and 
thus  the  final  magnetizing  force  is  the  maximum  that  has  been 
used.  This  is  better  than  ending  with  a  weak  value  of  H,  be- 
cause under  weak  fields  the  iron  will  not  come  to  its  final  mag- 
netization as  promptly  as  under  stronger  fields. 

The  remaining  portion  of  the  curve,  PJA,  can  be  determined 
in  precisely  the  same  way.  With  K  and  S  set  so  that  the  iron 
is  at  P,  let  the  connections  from  S  to  the  ring  Z  be  inter- 
changed. This  will  carry  the  iron  to  A.  By  turning  Fig.  76 
bottom  side  up  it  will  be  seen  that  the  curve  PJA  corresponds 
exactly  with  the  former  curve  A  DP;  and  it  can  be  traced  by 
repeating  the  observations  previously  made,  but  the  ordinates 
must  now  be  plotted  from  the  line  AN.  Or,  since  the  curves 
are  the  same,  the  former  set  of  ordinates  can  be  used  again  for 
the  second  portion  of  the  curve. 

The  actual  plotting  of  the  curve  AJP  downward  from  the 
line  AN  is  often  awkward,  as  usually  AN  does  not  coincide 
with  one  of  the  main  divisions  of  the  cross-section  paper.  An 
easier  way  is  to  lay  a  second  piece  of  paper  under  the  curve 
A  DP  and  fasten  both  together  by  two  pins  through  A  and  P 
Several  points  along  the  path  of  the  curve  are  pricked  through 
both  papers  with  a  needle  point.  The  lower  paper  is  then 
placed  over  the  curve,  with  the  points  A  and  P  interchanged, 
and  fastened  on  the  two  pins.  The  intermediate  points  are 
then  pricked  through  on  to  the  other  sheet,  thus  outlining  the 
curve  PJA. 

159.  Determination  of  the  Values  of  B. — In  this  experiment 
the  change  in  the  magnetic  induction  is  not  a  reversal,  but  a 
change  from  ea,  Fig.  76,  in  one  direction  to  eb  in  the  other. 
This  gives  a  total  change  of  ab,  which  may  be  denoted  by  the 
symbol  AB.  The  corresponding  change  in  flux  turns  in  the 
secondary  circuit  is  ABAn,  and  this  is  measured  by  the  gal- 
vanometer deflection,  the  relation  being, 

ABAn  =  cd  (1) 


MAGNETIC  TESTS  OF  IRON  AND  STEEL  185 

The  change  in  the  magnetic  induction  is  then 

AB  =  ~L d  =  Jd>  (2) 

and  this  is  plotted  as  the  ordinate  ab,  Fig.  76,  the  values  being 
laid  off  from  the  axis  OH.  After  the  curve  is  drawn  the  true 
axis,  FE,  is  drawn  through  the  middle  of  the  figure. 

160.  The  magnetic  ballistic  constant,  c,  may  be  determined 
by  means  of  a  known  mutual  inductance  as  in  the  previous 
methods.     From  Article  151, 

MF  =  cd'. 

where  /'  is  the  change  in  the  current  through  the  primary  of 
the  inductance  M,  measured  in  C.G.S.  units.  Since  c—  JAn, 
from  (2),  the  deflection  of  the  galvanometer  for  any  desired 
value  of  J  is, 

A'  -  MI', 

-JAn 

Usually  it  is  convenient  to  choose  J  =  100,  and  then  adjust 
the  resistance  in  series  with  the  galvanometer  until  the  deflec- 
tion is  dr  for  a  change  of  /'  C.G.S.  units  of  current  in  the  pri- 
mary of  the  calibration  coil.1 

The  secondary  of  this  mutual  induction  must  remain  a  part 
of  the  galvanometer  circuit,  as  shown  at  M ,  Fig.  77,  since  any 
change  of  the  resistance  of  this  circuit  will  change  the  value  of 
the  constant. 

161.  The  values  of  the  magnetizing  force,  H,  are  determined 
by  the  current,  which  may  be  read  by  an  ammeter.     Values 
of  H  may  then  be  computed  as  shown  in  Articles  153  or  155. 

1  NOTE. — Inasmuch  as  no  current  is  used  for  the  ring  Z  while  the  con- 
stant of  the  galvanometer  is  being  determined,  the  same  battery,  am- 
meter, and  rheostats  can  be  used  to  furnish  the  current  /'.  To  make 
this  change  it  is  only  necessary  to  exchange  the  connections  pq  for  p'q' 
Fig.  77. 

Since  the  zero  position  of  a  sensitive  D'Arsonval  galvanometer  depends 
upon  the  direction  in  which  it  was  last  deflected,  all  deflections  should 
be  in  one  direction  only,  and  therefore  the  constant  must  be  determined 
for  deflections  in  this  direction  also.  If  greater  accuracy  is  desired  the 
scale  should  be  calibrated  by  determining  the  value  of  J  for  deflections 
throughout  the  range  that  will  be  used. 


186  ELECTRICAL  MEASUREMENTS 

These  values  are  plotted   along  the  axis  FCE,  with  C  as 
the  origin. 

162.  Energy  Loss  through  Hysteresis. — Since  it  requires  a 
reversed  current  to  bring  the  magnetization  of  a  ring  or  bar  to 
zero,  there  is  always  a  considerable  loss  of  energy  when  a  piece 
of  iron  is  carried  through  a  cycle  of  magnetic  changes.  This 
energy  is  represented  by  the  area  of  the  hysteresis  loop, 
which  is  narrow  for  wrought  iron  while  for  cast  iron  it  is  large 
and  broad. 

The  amount  of  energy  thus  transformed  into  heat  can  be 
determined  as  follows:  The  work  required  to  magnetize  the 
iron  is  partly  lost  as  heat  in  the  magnetizing  solenoid.  This  is 
the  regular  Ri2  loss.  Another  part  is  used  in  maintaining  the 
current  against  the  induced  E.M.F.  due  to  the  newly  formed 
magnetic  field.  This  induced  E.M.F.  is 

d*  ANdB, 

^  dt   "  dt 

where  A  is  the  cross  section  of  the  iron  and  N  is  the  number  of 
turns  in  the  magnetizing  solenoid. 

Since  e  is  negative,  i.e.,  opposed  to  the  current,  the  positive 
work  required  to  maintain  the  current  is 


C  CANidB  7         CAHI  7:        Volume  ("TT ._ 

Jf  =    J  CTYft  =    j  -  -,-T—d*  =    I  — j— dB  =  — r I  HdB 

J  J        dt  J     4*  4;r     J 

since  #7  =  4.irNi. 

For  one  complete  cycle  J  HdB  is  the  area  of  the  hysteresis 

curve,  measured  in  the  units  of  H  and  B.  This  area,  then, 
gives  the  energy  expended  per  cycle  in  each  12.57  cc.  of  iron. 
163.  Permanent  Magnets. — Referring  to  the  hysteresis 
curve  for  the  bar  and  yoke,  it  is  seen  that  the  bar  is  still  mag- 
netized after  the  current  is  stopped.  When  the  bar  is  with- 
drawn from  the  yoke,  it  is  in  the  same  condition  as  before. 
The  bar  itself  is  a  permanent  magnet,  and  the  ends  where 
the  flux  enters  and  leaves  the  iron  are  its  poles. 


CHAPTER  XIII 


ELECTROMAGNETIC  INDUCTION 

164.  Electromagnetic  Induction. — When  the  current  flowing 
through  a  circuit  is  started  or  stopped,  or  changed  in  any  man- 
ner whatever,  it  is  observed  that  other  currents  are  set  up  in  all 
of  the  other  closed  circuits  which  are  near  the  first  one.  If 
some  of  these  circuits  are  not  closed,  the  tendency  to  produce  a 
current  is  present  just  the  same,  but  being  open  circuits  no 
current  results.  In  other  words,  an  E.M.F.  is  induced  in  every 
conductor  near  a  circuit  in  which  the  current  is  varying.  If  a 
current  in  the  second  circuit  is  varying,  there  will  be  a  corre- 
sponding E.M.F.  induced  in  the  first  circuit.  This  action  is 
called  mutual  induction. 


FIG.  78. — Mutual  induction. 

165.  Laws  of  Mutual  Induction. — A  very  satisfactory  study 
of  mutual  induction  can  be  made  with  a  pair  of  coils  and  a 
sensitive  ballistic  galvanometer,  connected  as  shown  in  Fig.  78. 
One  of  the  coils  is  joined  in  series  with  a  battery,  an  ammeter,  a 
key,  and  a  rheostat,  r,  for  varying  the  current  over  a  wide 
range.  This  is  called  the  primary  circuit,  and  whichever  coil 

187 


188  ELECTRICAL  MEASUREMENTS 

is  used  in  this  circuit  is  called  the  primary  coil.  The  other  coil, 
called  the  secondary,  is  connected  in  series  with  the  galvanom- 
eter and  a  resistance  box.  In  order  that  the  current  may  be 
varied  without  affecting  the  galvanometer  when  a  reading  is 
not  desired,  a  switch  or  key,  Kf  will  be  required  in  the 
secondary  circuit. 

It  is  also  desirable  to  open  the  secondary  circuit  before 
making  the  reverse  change  in  the  primary  current,  in  order  to 
avoid  deflecting  the  galvanometer  to  the  other  side  of  its  rest- 
ing point.  There  may  be  a  shift  of  the  zero,  or  resting,  point 
of  the  galvanometer  after  it  has  been  deflected  in  the  opposite 
direction.  In  this  case  the  first  two  or  three  readings  in  either 
direction  should  be  discarded.  In  taking  a  series  of  readings 
it  is  best  to  keep  the  deflections  in  the  same  direction  and 
never  allow  the  galvanometer  to  swing  across  zero  to  the  other 
side.  If  this  precaution  is  taken  the  zero  position  should  be 
quite  constant. 

166.  The  Effect  of  Varying  the  Primary  Current.— The 
first  part  of  this  experiment  is  to  study  the  relation  between  the 
change  in  the  primary  current  and  the  resulting  deflection 
of  the  galvanometer.  The  primary  current  is  adjusted  to  such 
a  value  that  closing  K",  with  Kf  closed  of  course,  will  give  a 
fairly  large  deflection.  Note  the  effect  of  opening  K" ';  also 
the  effect  of  closing  K"  first,  and  then  closing  Kf. 

The  actual  value  of  the  current  has  no  effect  upon  the  deflec- 
tion, as  this  is  the  same  when  the  current  is  changed  from  zero 
to  one  ampere  as  when  the  change  is  from  four  amperes  to 
five  amperes.  If  a  primary  current  of  one  ampere  is  reversed, 
the  change  in  the  current  is  evidently  two  amperes,  and  the 
deflection  will  be  twice  as  great  as  for  either  making  or  breaking 
the  circuit. 

Law  I. — The  quantity  of  electricity  flowing  through  the 
secondary  circuit  depends  directly  upon  the  change  produced 
in  the  primary  current. 

To  investigate  this  relation,  the  rheostat  r  is  adjusted  to  give 
the  largest  current  that  is  to  be  used,  and  R  and  P  are  set  so  as 


ELECTROMAGNETIC  INDUCTION  189 

to  make  the  corresponding  fling  of  the  galvanometer  as  large 
as  can  be  conveniently  measured.  Starting  with  this  current, 
the  deflection  is  read  when  the  circuit  is  closed.  The  reverse 
kick  of  the  galvanometer  is  avoided  by  opening  K'  as  soon  as 
the  reading  is  obtained,  and  then  opening  K".  This  reading 
should  be  repeated  to  make  sure  of  consistent  results.  The 
current  should  be  kept  flowing  as  little  as  possible  to  avoid 
heating  the  coils — especially  changing  the  resistance  of  the 
secondary  coil  by  warming  it.  The  current  is  now  reduced 
about  10  per  cent,  and  another  set  of  readings  obtained,  and 
so  on  until  the  current  is  reduced  to  zero. 

Since  the  scale  readings  are  proportional  to  tan  26,  it  may 
be  necessary  to  correct  them  by  the  use  of  a  calibration  curve. 
The  corrected  readings  are  then  plotted  as  ordinates  against 
the  corresponding  changes  in  the  primary  current.  This 
should  be  a  straight  line  passing  through  the  origin,  and  repre- 
sented by  the  equation, 

d  =  al,  (1) 

where  a  is  the  slope  of  the  curve.  The  value  of  a  depends  upon 
the  number  of  turns  of  wire  in  the  primary  and  secondary 
coils,  and  it  also  contains  as  one  factor  the  reciprocal  of  the 
resistance  of  the  secondary  circuit. 

167.  The  Effect  of  Varying  the  Secondary  Resistance.— The 
second  part  of  the  experiment  deals  with  the  effect  of  changing 
the  secondary  circuit.  Keeping  the  primary  circuit  constant, 
so  that  there  will  be  the  same  change  in  the  current  each  time 
the  key  is  closed,  a  series  of  deflections  are  obtained  by  using 
different  resistances  in  the  secondary  circuit.  At  each  step 
two  or  three  readings  are  taken  in  the  same  way  as  before.  All 
of  the  resistance  in  this  circuit  should  be  measured  under  the 
conditions  of  the  experiment,  especially  if  the  coil  has  been 
warmed  appreciably  by  the  current  in  the  adjacent  primary. 

The  galvanometer  should  not  be  joined  directly  in  series  with 
the  rest  of  the  circuit,  for  when  the  resistance  of  this  circuit  is 
varied  it  will  alter  the  damping  factor  of  the  galvanometer,  and 


190 


ELECTRICAL  MEASUREMENTS 


therefore  its  deflections  will  not  be  proportional  to  the  quanti- 
ties of  electricity  that  are  discharged  through  it.     This  trouble 
can  be  avoided  by  using  a  constant  damping  shunt  consisting 
of  two  resistances,  P  and  Q.    P  should  be  a  few  ohms,  as 
many  as  necessary,  and  P  +  Q  sufficiently  large  to  give  critical 
damping  to  the  galvanometer,  that  is,  as  large  as  possible  and 
still  keep  the  swing  of  the  mirror  aperiodic.     A  universal  shunt 
of  the  proper  resistance  may  conveniently  replace  P  +  Q. 
The  total  resistance  of  the  secondary  circuit  is 
R'  =  R  +  h  +  p, 

where  h  denotes  the  resistance  of  the  secondary  coil,  p  the  com- 
bined resistance  of  the  galvanometer  and  shunt,  and  R  is  the 
additional  resistance  in  the  box  R. 


Q'  o  R 

FIG.  79. — Laws  of  mutual  induction. 

It  will  be  found  that  the  deflections  become  larger  as  R'  is 
made  smaller  by  decreasing  the  resistance  in  R\  and  plotting 
the  corrected  deflections  against  l/R'  will  give  a  straight  line 
passing  through  the  origin,  as  in  the  first  part.  Inasmuch, 
however,  as  R  is  the  only  observed  part  of  R',  and  h  +  p  may 
be  unknown,  it  is  more  satisfactory  to  plot  l/d  as  ordinates 
against  the  corresponding  values  of  R.  This  will  give  a 
straight  line  as  before,  but  not  through  the  origin.  Extending 
this  back  to  the  axis  of  R  it  will  intersect  it  at  a  point  h  +  p 
ohms  behind  the  origin  of  R:  hence  this  point  is  the  origin  of 
R'. 


ELECTROMAGNETIC  INDUCTION  191 

The  relation  between  d  and  R'  is  given  then  by  the  expres- 
sion, 

i=6J?',orrf=^  (2) 

where  R'  is  the  total  resistance  in  the  secondary  circuit  and 
6  is  the  slope  of  the  line  O'B.  The  value  of  b  depends  upon  the 
number  of  turns  in  each  of  the  coils,  and  it  also  contains  as  one 
factor  the  constant  value  of  the  current  which  was  made  and 
broken  in  the  primary  circuit  when  the  deflections  were 
observed. 

Law  II. — The  quantity  of  electricity  flowing  through  the 
secondary  circuit  varies  inversely  as  the  resistance  of  the 
circuit. 

From  these  two  relations  it  is  seen  that  the  complete  expres- 
sion for  the  relation  between  the  deflection,  the  change  in  the 
primary  current,  and  the  resistance  of  the  secondary  circuit 
is 

d  =  f  ^7  (3) 

where  /  is  the  proportionality  factor.  Evidently  f/Rf  is  the  a 
of  eq.  (1),  if  Rr  is  the  constant  resistance  of  the  secondary 
circuit  when  curve  A  was  determined.  Likewise,  //  is  the 
6  of  Eq.  (2)  if  /  denotes  the  change  that  was  made  in  the 
primary  current  when  curve  B  was  determined. 

168.  Meaning  of  Mutual  Inductance. — The  total  quantity 
of  electricity  discharged  through  the  galvanometer  is,  then, 

Q  =  kd  =kf^  (4) 

where  k  is  the  usual  ballistic  constant. 

The  ballistic  galvanometer  indicates  the  passage  of  a  quan- 
tity of  electricity  through  it,  and  from  the  nature  of  the  circuit 
to  which  it  is  connected  it  is  seen  that  this  quantity  does  not 
come  from  the  discharge  of  a  condenser,  but  represents  the 


192  ELECTRICAL  MEASUREMENTS 

passage  of  a  transient  current.  And  furthermore,  if  there  is  a 
current  in  the  secondary  circuit  there  must  be  an  E.M.F.  caus- 
ing this  current. 

On  page  167  it  was  shown  that  an  E.M.F.  is  induced  in  a 
wire  which  cuts  across  a  magnetic  flux.  In  the  present  case,  as 
well  as  in  the  experiments  of  the  preceding  chapter,  the  wire  is 
stationary  while  the  flux  cuts  across  it.  When  this  wire  is 
wound  into  a  coil  of  n  turns  the  flux  cuts  the  wire  in  each 
turn,  thus  inducing  an  E.M.F.  in  each  turn.  It  should  there- 
fore be  possible  to  express  the  value  of  the  E.M.F.  in  the  sec- 
ondary coil  in  a  form  similar  to  Eq.  (6)  in  Art.  138.  The 
total  E.M.F.  induced  in  the  coil  is,  then, 

.-.*  (5) 

where  <j>  denotes  the  flux  linked  with  the  n  turns  of  the  coil. 
This  is  the  E.M.F.  causing  the  current  through  the  galva- 
nometer and  the  secondary  circuit.  In  case  the  same  flux  does 
not  pass  through  each  turn  of  the  coil  the  two  factors,  <£  and  n, 
of  the  "  flux-turns"  cannot  be  separated,  but  this  quantity,  4>n, 
must  then  be  considered  as  a  summation  extending  to  all  the 
turns  of  the  coil. 

The  two  coils,  T  and  'S,  Fig.  78,  are  simMar  to  the  primary  and 
secondary  windings  on  the  iron  ring  Z,  Fig.  77,  except  that  now 
the  magnetic  circuit  is  wholly  of  air.  The  M.M.F.  due  to  the 
current  in  T  is  4arNi',  and  this  causes  a  flux, 


Reluctance 

through  this  coil  and  the  surrounding  air.  The  portion,  0,  of 
this  flux  passing  through  the  n  turns  of  coil  S  depends  upon  the 
relative  position  of  the  two  coils.  Let  p  denote  this  fraction. 
Then  <f>  =  p$,  and  the  E.M.F.  induced  in  this  coil  by  the 
change  in  the  primary  current  is, 

_        d^  /4irNif\  _      4irpNn     <W_  ^ 

dT  ~  pndt\ReLl  ~  Reluctance  dt 


. 

ELECTROMAGNETIC  INDUCTION  193 

Writing  a  single  symbol  for  these  various  constants, 

.-*£  (7) 

where  the  coefficient  M  is  called  the  coefficient  of  mutual  in- 
duction, or  simply  the  mutual  inductance  of  the  pair  of  coils. 

Definition.  —  One  henry  of  mutual  inductance  is  the  induc- 
tance between  two  circuits  when  an  E.M.F.  of  one  volt  is  pro- 
duced in  one  of  them  when  the  inducing  current  in  the  other 
changes  at  the  rate  of  one  ampere  per  second. 

Strictly  speaking,  eq.  (7)  should  be  written  —  e,  for  if  atten- 
tion is  paid  to  the  direction  of  e  when  the  current  is  increased, 
that  is  when  dl  is  positive,  it  is  found  to  be  directed  round  the 
coil  in  the  opposite  sense  to  the  current,  /.  If  we  are  looking 
only  for  the  numerical  values  of  e  and  M  the  sign  does  not 
matter,  but  in  case  the  direction  of  the  induced  current  is 
considered  it  is  necessary  to  write, 


i68A.  Value  of  the  Mutual  Inductance.  —  When  a  vary- 
ing current  flows  through  a  circuit  which  is  wound  into 
a  coil,  like  the  primary  and  secondary  coils  in  the  present 
case,  it  is  necessary  to  modify  the  statement  of  Ohm's 
law  as  given  for  steady  currents.  The  E.M.F.  necessary  to 
maintain  a  varying  current  is, 

e  =  Ri  +  L  -^> 

where  Ri  is  the  E.M.F.  necessary  to  maintain  the  current 
through  the  resistance  R,  and  L  di/dt  is  the  E.M.F.  required 
to  make  the  current  change  by  an  amount  di  in  the  time  dt, 
because  of  the  self  inductance,  L,  of  the  circuit. 

1  It  is  necessary  to  add  this  term  in  order  to  keep  the  equation  correct. 
It  drops  out  upon  integration  and  does  not  affect  the  final  result.  The 
full  meaning  of  this  term  is  explained  in  the  proper  place  a  few  pages 
below.  (See  eq.  (5),  Article  172.) 

13' 


194  ELECTRICAL  MEASUREMENTS 

Integrating  this  equation  with  respect  to  the  time, 
/v  r?  rt'  rq=Q         /^=o 

\  edt  =    I  Ridt  +    I  Ldi  =  R    \  dq  +  L    \  di  =  RQ  (5) 

Jt  Jt  Jt  Jq  =  0  Ji  =  Q 

•where  the  integration  is  extended  from  the  time,  t,  before  the 
primary  current  begins  to  flow  till  the  time,  t't  when  the 
primary  current  has  reached  its  steady  value,  7. 
But  from  (7), 

j     edt  =   \Mdi'  =  MI, 

and    therefore, 

MI  =  PQ,  (4') 

or,  solving  for  the  value  of  the  mutual  inductance, 

(8) 

If  I  is  expressed  in  amperes,  R  in  ohms,  and  Q  in  coulombs, 
then  M  will  be  given  in  henries. 

169.  Calculation  of  the  Value  of  M. — Returning  now  to  the 
consideration  of  the  first  part  of  this  experiment,  let  R'  denote 
the  fixed  value  of  the  resistance  of  the  entire  secondary  circuit. 
Then  from  (8), 

M  =  R'  Q  =  R'  y  =  R'ka,  (9) 

where  each  symbol  has  the  meaning  that  has  been  assigned 
above.  The  value  of  k  can  be  determined  as  shown  below, 
while  R'  and  a  are  obtained  from  the  curves. 

From  the  second  part  of  the  experiment,  in  which  the  pri- 
mary current  was  kept  constant  at  /'  amperes, 

M_QR'_.kdR'       fc 

M  -        p  —p~     =    kJT  UW 

.  The  constant  b  is  the  slope  of  the  curve  O'B,  and  k  is  deter- 
mined as  follows. 


ELECTROMAGNETIC  INDUCTION 


195 


170.  To  Determine  the  Constant  of  the  Ballistic  Galvan- 
ometer.— When  the  constant  of  a  ballistic  galvanometer  is 


(G) 1 


FIG.  80. — To  determine  the  constant  of  G. 

determined  by  the  usual  method  with  a  condenser  and  standard 
cell  (see  Article  32)  the  galvanometer  is  used  on  open  circuit. 
When  it  is  used  on  a  closed  circuit,  as  in  the  case  above,  the 
damping  is  very  much  greater,  and  the  same  quantity  of  elec- 
tricity discharged  through  the  galvanometer  will  produce  a 
much  smaller  deflection.  Therefore  the  constant  should  be 
determined  with  the  galvanometer  under  the  conditions  in 
which  it  has  been  used.  This  requires  that  the  secondary 
circuit  be  opened  long  enough  to  discharge  the  condenser 
through  it,  and  then  immediately  closed  again.  This  can  be 
done  by  using  two  keys,  as  shown  in  Fig.  80,  or  better  by  a 
double  testing  key  in  which  all  the  contacts  are  made  during  a 
single  motion  of  the  hand. 

This  method  for  determining  the  constant  of  the  galvanom- 
eter is  not  recommended  where  much  accuracy  is  required  and 
it  should  be  resorted  to  only  when  no  other  method  is  available. 
In  theory  it  is  simple,  but  owing  to  the  increased  damping  of 
the  galvanometer  when  connected  to  a  closed  circuit  it  is 
difficult  to  determine  just  what  value  of  d  should  be  used  for 
computing  k.  The  following  method  is  more  satisfactory 
and  gives  more  accurate  results. 


196  ELECTRICAL  MEASUREMENTS 

171.  Constant  of  a  Ballistic  Galvanometer  by  Means  of  a 
Coil  of  known  Mutual  Inductance.  —  The  foregoing  experi- 
ment furnishes  a  method  for  obtaining  the  constant  of  a 
ballistic  galvanometer  when  a  known  mutual  inductance  is  at 
hand.  For  let  the  secondary  of  such  a  pair  of  coils  be  included 
as  a  part  of  the  galvanometer  circuit.  When  not  in  use  it  will 
not  affect  the  galvanometer  except  as  so  much  resistance  in  the 
circuit,  and  when  it  is  desired  to  find  the  constant  k,  it  is  only 
necessary  to  pass  a  current  through  the  primary  and  observe 
the  galvanometer  deflection  when  this  current  is  made  or 
broken. 

From  eq.  (9)  we  have, 

MI  =  R'kd,          or,  k 


This  use  of  a  known  mutual  inductance  is  especially  useful 
in  cases  where  it  is  desired  to  obtain  the  constant  of  the  galva- 
nometer without  opening  the  galvanometer  circuit.  In  fact  it 
would  be  useful  in  the  above  experiment  if  the  use  of  a  mutual 
inductance  is  thoroughly  understood  by  the  student. 

1  7  1  A.  Secondary  Definition  of  Mutual  Inductance.  —  In  the  ex- 
periments of  Chapter  XII,  a  different  constant  was  required,  viz., 
the  magnetic  ballistic  constant,  c,  whch  was  used  in  the  relation, 

</>n  =  cd  (1) 

where  d  is  the  deflection  due  to  a  change  of  <f>n  flux-turns 
through  the  test  coil. 

The  E.M.F.  induced  in  this  case  was,  (see  Article  138). 

.--*$  (2) 

In  the  case  above  (see  Article  168),  the  E.M.F.  induced  in 
the  secondary  coil  was, 


where  /  was  the  primary  current. 


ELECTROMAGNETIC  INDUCTION  197 

Equating  (2)  and  (3),  and  integrating,  gives, 

<j>n  =  MI  (4) 

Therefore,  when  a  current  7  is  flowing  through  the  primary 
of  a  pair  of  coils  whose  mutual  induction  is  M ,  there  are  <f>n 
flux-turns  through  the  secondary,  and  it  is  the  changing  of 
this  flux  that  causes  the  induced  E.M.F.  in  the  secondary. 
If  the  secondary  is  so  arranged  that  all  of  the  flux  passes 
through  each  of  its  n  turns,  then  <£  is  the  value  of  the  actual 
flux.  In  case  some  of  the  flux  does  not  pass  through  all  of  the 
turns,  this  product  is  to  be  considered  as  a  summation,  counting 
the  actual  flux  each  time  it  passes  through  a  turn  of  the 
secondary.  In  either  case  the  value  of  <j>n  is  determined  by 
measuring  M ,  and  therefore  the  question  whether  all  the  flux 
passes  through  all  the  turns  of  the  secondary  is  of  no  moment. 

Evidently  all  of  the  quantities  in  Eq.  (4)  must  be  measured 
in  C.G.S.  units. 

172.  Meaning  of  Self  Inductance. — It  is  but  a  step  from 
considering  the  action  of  a  current  on  an  adjacent  circuit  to  the 
case  of  action  upon  the  same  circuit.  If  the  second  circuit 
above  were  included  as  a  part  of  the  primary  circuit  then  the 
induced  E.M.F.  would  be  in  the  same  circuit  as  the  inducing 
current,  and  in  a  direction  opposed  to  it.  Furthermore,  this 
same  kind  of  action  would  appear  in  every  turn  of  the  primary 
coil,  the  current  in  each  turn  inducing  an  E.M.F.  in  each  of  the 
other  turns.  The  total  E.M.F.  induced  in  the  circuit  by  the 
inducing  current  is, 

Tdi 
e  =  ~Ldt 

where  the  —  sign  means  the  same  as  before  and  L  is  a  constant 
depending  upon  the  number  of  turns  and  the  dimensions  of  the 
circuit.  It  is  called  the  Self  Inductance  of  the  circuit  and  is 
measured  in  henrys. 

Definition. — One  henry  of  self  inductance  is  the  inductance  in 
a  circuit  when  an  electromotive  force  of  one  volt  is  produced 
by  the  current  changing  at  the  rate  of  one  ampere  per  second. 


198  ELECTRICAL  MEASUREMENTS 

173.  Starting  and  Stopping  a  Current.  —  When  an  E.M.F.,  E., 
is  applied  to  a  circuit,  the  current  starts  at  the  value  zero  and 
increases  to  its  final  steady  value.  While  the  current  is  thus 
increasing  there  is  induced  in  the  same  wire  an  E.M.F.  which 
is  opposed  to  the  current.  The  value  of  this  E.M.F.  is 

Tdi 

e=~Ldt 
The  total  E.M.F.  in  the  circuit  at  any  instant  is,  therefore, 

E  +  e  =  E-L^t  (2) 

and  by  Ohm's  law  the  current  at  the  same  instant  will  be, 

T-T         di 

,  _  E^I*  (3) 

R 

At  the  start,  when  the  current  is  changing  most  rapidly, 
the  second  term  in  the  numerator  is  nearly  equal  to  E.  As  the 
current  approaches  its  final  value  it  changes  more  slowly,  the 
counter  E.M.F.  becomes  small,  and  finally  the  steady  value 
of  the  current  is 


Equation  (3)  is  often  written  in  the  form 


(4) 


(5) 


which  shows  that  while  a  part,  Ri,  of  the  applied  E.M.F.  is 

di 
effective  in  maintaining  the  current  i,  another  part,  +  L  -T-> 

is  required  to  make  the  current  increase.     This  equation  is 
very  important  in  all  discussion  of  varying  currents. 

174.  Dying  Away  of  a  Current.  —  Suppose  that  a  steady  cur- 
rent has  been  flowing  in  a  circuit  when  suddenly  the  battery, 
or  other  source  of  E.M.F.  is  removed  without  breaking  the 
circuit  or  in  any  way  changing  its  resistance  or  inductance. 
Eq.  (5)  now  becomes 


. 

ELECTROMAGNETIC  INDUCTION  199 

0  =  Ri  +  L  -r.  (6) 

and  the  current  thus  left  to  itself  dies  away  to  zero,  as  is 
shown  by  the  following  relations. 

Rewriting  (6)  to  separate  the  variables, 

-  =         ~dt  (7) 

1  L 

The  integral  of  this  is 

tog*  =  --  f  +  c  (8) 

or  in  the  exponential  form, 

_Rt 

i=Ce      L  (9) 

The  value  of  the  constant  of  integration,  C,  is  given  by  the  fact 
that  at  the  start,  when  t  =  0,  the  current  had  the  value  J. 
Putting  these  values  in  (9)  gives,  I  =  C.  Therefore  (9) 
becomes 

i  -/«-r  (10) 

From  this  equation  it  appears  that  it  is  the  self  inductance  that 
prevents  the  current  from  falling  to  zero  immediately,  and  the 
greater  the  self  inductance  the  more  slowly  will  the  current  die 
away.  If  it  is  desired  to  state  how  rapidly  the  current 
decreases  it  is  necessary  to  state  how  long  it  takes  for  the  cur- 
rent to  fall  to  half  value — or  to  some  other  definite  fraction  of 
its  original  amount — since  the  value  of  i  from  (10)  will  reach 
zero  only  after  an  infinite  time. 

When  t  has   the   value   p,  the  current  equals  -,  or  ^T,, 

and  this  interval  in  which  the  current  falls  to  0.368  of  its 
original  value  is  called  the  time  constant  of  the  circuit. 

175.  Beginning  of  a  Current. — When  a  circuit  containing  an 
E.M.F.  is  closed  the  current  rises  from  zero  to  its  final  value  at 
a  rate  depending  upon  the  self  inductance  in  the  circuit.  This 
rate  of  increase  can  be  found  from  the  equation 


200  ELECTRICAL  MEASUREMENTS 

E  -  Ri  +  L|- 
Separating  the  variables  gives  this  in  the  form 


jj- 

the  integral  of  which  is 

/#  #*   , 

log  (g  -  t)  =      -  ^  +  c, 

or, 


The  constant  of  integration  is  determined  by  the  condition 
that  when  t  =  0,  i  =  0.  Hence,  C  =  E/R  =  7,  where  /  is 
the  final  value  of  the  current. 

Therefore, 

m 
i  =  I  -  It  ~  T- 

Here  also  it  is  seen  that  it  is  the  self  inductance  that  keeps 
the  current  from  rising  suddenly  to  its  full  value  as  soon  as 
the  circuit  is  closed.  The  greater  the  self  inductance  in  com- 
parison with  the  resistance,  the  more  slowly  will  the  current 
rise,  but  it  never  quite  reaches  its  maximum  value.  Therefore 
in  comparing  different  circuits  it  is  necessary  to  compare  the 
periods  taken  for  the  currents  to  rise  to  half  value  —  or  to  some 
other  definite  fraction  of  the  final  steady  values.  When 

t  =  j>  the  current  lacks  —of  its  final  value,    and   this   per- 

iod in  which  the  current  rises  to  0.632  of  its  maximum  value  is 
called  the  time  constant  of  the  circuit. 

^"Problem  A.  —  Given  a  circuit  in  which  the  resistance  is  10  ohms, 
and  the  self  inductance  is  .01  henry,  draw  a  curve  showing  the  rise  of 
the  current  for  the  first  0.004  of  a  second  after  applying  a  steady 
E.M.F.  of  100  volts. 


ELECTROMAGNETIC  INDUCTION 


201 


Problem  B. — The  above  circuit  is  placed  in  parallel  with  a  non- 
inductive  resistance  of  10  ohms,  when  the  current  in  the  main  line  is 
20  amperes.  Draw  a  curve  showing  how  the  current  dies  away  hi 
the  parallel  circuits  when  the  main  line  switch  is  suddenly  opened. 

The  logarithms  which  appear  in  the  equations  through  the 
process  of  integration  are,  of  course,  not  the  common  logarithms 
with  the  base  10,  but  are  the  natural  logarithms  with  the  base 
c  =  2.718  +.  Therefore  in  finding  the  time  at  which  the 
current  will  have  a  given  fraction  of  its  maximum  value, 
the  following  table  will  be  useful. 


•»T               1_ 

Natural  logarithms. 

Number 

Tabular  value 

Numerical  value 

0.9 

9.895-10 

-0.105 

0.8 

9.777-10 

-0.223 

0.7 

9.643-10 

-0.357 

0.6 

9.489-10 

-0.511 

0.5 

9.307-10 

-0.693 

0.4 

9.084-10 

-0.916 

0.3 

8.796-19 

-1.204 

0.2 

8.391-10 

-1.609 

0.1 

7.697-10 

-2.303 

0.07 

7.341-10 

-2.659 

0.04 

6.781-10 

-3.219 

0.01 

5.395-10 

-4.605 

CHAPTER  XIV 


MEASUREMENT  OF  SELF  AND  MUTUAL  INDUCTANCE 

176.  Comparison  of  Two  Self  Inductances. — The  self  in- 
ductance of  a  coil  can  be  measured  by  the  bridge  method, 
using  variable  currents.  A  Wheatstone  bridge  arrangement 
is  set  up  as  shown  in  Fig.  81,  with  the  coil  to  be  measured 
as  one  arm  of  the  bridge  and  a  variable  standard  of  self  in- 
ductance in  the  corresponding  arm. 
The  other  two  arms  consist  of  two 
non-inductive  resistance  boxes 
which  can  be  adjusted  to  balance 
the  bridge.  A  galvanometer  G  and 
a  battery  B  are  connected  in  the 
usual  way  with  the  double  key  Kr, 
R". 

In  order  to  use  variable  currents 
the  secondary  of  an  induction  coil, 
or  other  source  of  alternating  cur- 
rent, is  connected  to  the  bridge  at 
the  same  points  as  the  battery, 
without  disturbing  the  latter  con- 

8il7?nZKc°e°.0f  tW°  nections.  The  primary  of  this  in- 
duction  coil  is  connected  to  the  bat- 
tery through  the  switch  S".  The  same  battery  can  serve  in 
both  places  since  it  will  not  be  needed  in  one  circuit  while 
the  other  is  being  used.  As  an  ordinary  galvanometer  is  not 
deflected  by  an  alternating  current,  a  telephone  receiver  is 
connected  between  B  and  C  by  the  switch  £'.  The  two 
switches,  S'  and  S"  may  well  be  the  two  blades  of  a  double 
pole  switch  which  will  close  both  circuits  by  a  single  motion. 

202 


MEASUREMENT  OF  INDUCTANCE  203 

The  induction  coil  should  be  enclosed  in  a  well  padded  box 
to  reduce  the  noise  as  much  as  possible.  By  listening  at  the 
telephone  receiver  while  the  inductance  of  the  standard  is 
varied,  the  position  for  a  minimum  sound  is  readily  determined. 
Before  recording  the  readings  the  direct  current  balance  should 
again  be  tried  to  make  sure  that  the  resistances  have  not 
changed  while  the  second  balance  was  being  made.  The 
contacts  in  the  variable  standard  are  not  always  constant,  and 
perhaps  other  changes  may  occur.  A  Wheatstone  bridge  can 
be  balanced  for  varying  currents  only  when  both  the  resistances 
and  the  inductances  are  adjusted  to  the  proper  values. 

The  relation  between  the  inductances  and  resistances  which 
will  give  this  double  balance  may  be  found  as-follows:  Apply- 
ing Kirchhoffs  law  to  the  circuit  ABC  A  at  any  instant  when 
the  currents  are  i  and  if  gives 

Pi-  R'i'  -Lf^~  =  0  (1) 

similarly  for  BDCB, 

Qi-R"i'-L"^  =  Q  (2) 

since  there  is  no  current  from  B  to  C  as  indicated  by  no  sound 
in  the  telephone. 

Transposing  and  dividing  (1)  by  (2) 

U_   _  Pi  -  R'i'    _  P(i  -  R'  i'/P)  _  P 
L"  ~  Qi  -  R"i'  ~  Q(i  -  R"i'/Q)  ~  Q 


since  the  resistances  are  adjusted  for  a  direct  current  balance 
and  therefore  R'  /P  =  R"/Q. 

Thus  for  a  balance  with  varying  currents,  the  triple  equation 

V_  _R!_  _P 
L"  ~  R"  ~  Q 

must  be  satisfied. 

This  means  that  each  of  the  inductive  arms  of  the  bridge 
must  be  equally  inductive,  or  in  other  words,  the  henrys  per 


204 


ELECTRICAL  MEASUREMENTS 


ohm  must  be  the  same  for  each  arm  in  order  to  give  a  balance 
with  variable  currents.  All  of  this  is  on  the  supposition  that 
Lf  is  a  constant,  i.e.,  contains  no  iron  or  other  magnetic  sub- 
stance. It  is  seen  that  the  balance  is  independent  of  the 
particular  way  in  which  the  currents  are  made  to  vary. 

In  case  no  balance  can  be  obtained  within  the  range  of  the 
standard,  some  non-inductive  resistance  may  be  added  to  the 
more  inductive  of  the  two  inductive  arms.  And  by  varying 
this  resistance  several  readings  may  be  obtained  on  different 
parts  of  the  scale. 

An  Inductance  Bridge 

An  inductance  bridge  box  is  made  by  Leeds  and  Northrup, 
and  in  general  appearance  resembles  the  Wheatstone  bridge 


FIG.  81a. 

box  made  by  the  same  firm.  The  ratio  arms  are  similar  in  the 
two  boxes  and  each  has  a  rheostat  arm,  but  as  the  actual  re- 
sistance in  this  arm  is  immaterial  in  the  inductance  bridge  the 
fine  adjustment  is  made  by  two  small  circular  rheostats,  each 
having  many  steps.  The  ratio  arms  correspond  to  P  and  Q 


MEASUREMENT  OF  INDUCTANCE  205 

of  Fig.  81,  while  the  rheostat  arm  of  variable  resistance  is 
connected  between  the  two  inductance  coils  as  shown  in  Fig. 
81a.  By  means  of  a  plug  key  the  galvanometer  connection  C 
may  be  joined  to  either  C'  or  C",  thus  putting  the  resistance 
R  in  series  with  either  R'  or  R"  as  desired.  The  resistance 
balance  is  easily  made  by  varying  R,  after  which  the  inductance 
balance  is  obtained  by  varying  L'.  The  value  of  the  unknown 
inductance  is,  then, 


where  Q/P  is  an  integer  power  of  10. 

177.  Comparison  of  a  Mutual  Inductance  with  a  Self  Induc- 
tance. —  The  preceding  method  gives  also  a  very  satisfactory 
way  of  measuring  the  mutual  inductance  between  two  coils. 
Call  the  self  inductances  of  the  coils  Lm  and  Ln,  and  their 
mutual  inductance  M.  The  E.M.F.  induced  in  these  coils 
when  connected  in  series  and  carrying  a  current  i,  is, 

' 


But  the  coefficient  of  di/dt  is  by  definition  the  inductance 
of  the  circuit,  and  if  the  two  coils  thus  joined  together  in 
series  were  made  one  arm  of  the  bridge  shown  in  Fig.  81,  the 
inductance  measured  by  that  method  would  be 

Ls  =  Lm  +  Ln  +  2M.  (1) 

If  one  of  the  coils  were  reversed,  the  E.M.F.  induced  in  each 
coil  by  the  mutual  inductance  of  the  other  would  be  reversed 
also,  and  therefore  the  inductance  measured  by  the  bridge  in 
this  case  would  be, 

Lr  =  Lm  +  Ln  -  2M  (2) 

Subtracting  (2)  from  (1)  gives 

La  -  Lr  = 


206 


ELECTRICAL  MEASUREMENTS 


Hence  to  determine  the  mutual  inductance  of  a  pair  of  coils 
it  is  only  necessary  to  measure  the  self  inductance  of  both  to- 
gether when  joined  in  series — first  with  the  coils  direct,  and 
again  with  one  of  the  coils  reversed.  One-fourth  of  the  differ- 
ence between  the  two  inductances  gives  the  value  of  the  mutual 
inductance. 

178.  Measurement  of  a  Self  Inductance  by  means  of  a 
Capacity.  Maxwell's  Method.— In  this  method  the  self 
inductance  to  be  measured  is  placed  in  one  arm  of  a  Wheat- 
stone  bridge,  the  other  arms  of  which  should  be  as  free  from 
inductance  as  possible.  By  closing  the  keys  in  the  usual  order 

the  bridge  can  be  balanced 
for  steady  currents  giving  the 
relation 

PS  =  QR         (i) 

But  even  though  the  bridge 
is  balanced,  if  the  galvan- 
ometer key  is  closed  first  there 
will  be  a  large  deflection  upon 
closing  the  battery  key.  This 
is  due  to  the  current  through 
the  inductive  branch  ABD 
not  being  able  to  reach  its  full 
value  as  quickly  as  the  cur- 
rent through  the  non-inductive 

branch  AED,  and  therefore  j  ust  at  the  start  the  fall  of  potential 
over  S  is  not  as  great  as  that  over  R.  This  effect  can  be  bal- 
anced by  a  suitable  inductance  in  P,  as  was  done  in  Article  176, 
or  by  placing  a  condenser  in  parallel  with  R  as  shown  in 
Fig.  82.  With  the  condenser  so  placed,  a  large  part  of  the 
current  through  P,  just  at  the  start,  will  flow  into  the  con- 
denser and  therefore  the  fall  of  potential  over  R  will  not  be  as 
large  as  it  would  be  without  the  condenser.  If  the  capacity 
of  the  condenser  is  of  the  proper  amount  this  latter  effect  will 
just  equal  the  effect  of  the  coil  in  reducing  the  current  through 


FIG.  82. — Comparison  of  capacity 
and   self   inductance. 


MEASUREMENT  OF  INDUCTANCE  207 

S,  and  therefore  the  potentials  of  B  and  E  will  rise  together. 
In  this  case  there  will  be  no  deflection  of  the  galvanometer 
even  though  its  key  is  closed  first. 

In  practice  then  the  bridge  is  first  balanced  for  steady 
currents.  Then  with  K%  closed  the  balance  is  tried  again.  If 
the  value  of  C  is  not  just  right  to  give  a  balance  with  varying 
currents,  and  if  it  cannot  be  varied,  its  effect  can  be  varied  by 
changing  R.  This  will  necessitate  a  corresponding  change  in  S 
and  another  trial  with  varying  currents  until  this  double 
balance  has  been  obtained.  A  telephone  in  place  of  the 
galvanometer  and  alternating  currents  can  be  used  equally 
well  for  the  second  balance.  Or  if  the  apparatus  is  at  hand,  it  is 
convenient  to  use  the  alternating  current  generator  with  a 
commutator  on  the  same  shaft  for  rectifying  the  galvanometer 
current.  The  140  volts  generated  by  the  machine  is  entirely 
too  much  to  use  directly  on  good  resistance  boxes.  By  means 
of  a  transformer  the  voltage  can  be  reduced  to  twelve  volts. 
The  secondary  can  be  connected  in  series  with  the  battery  of 
Fig.  82.  When  the  machine  is  not  running  the  direct  cur- 
rent balance  can  be  obtained  the  same  as  before.  When 
the  alternating  current  is  used  it  will  be  superimposed  on  the 
battery,  but  will  give  a  varying  current  all  the  same. 

When  no  current  flows  from  B  to  E  we  can  write  for  any  one 
instant, 

Pii  =  Qi*  +  L-^t  (2) 

and 

Ri  =  Si2  (3) 

where  n  =  i  +  i'  =  i  +  ^  =  i  +  ^  (CRi)=  i  +  CR~     (4) 

since  both  the  current  i  through  R,  and  if  flowing  into  the  con- 
denser are  added  together  to  make  the  current  i\  through  P. 
Substituting  in  (2)  the  value  of  iz  from  (3)  and  i\  from  (4) 
gives 

„  Ri  .    r  R  di 


208  ELECTRICAL  MEASUREMENTS 

or 

(PS  -  QR)i  +  PCRS  ^  =  LR  ~  (5) 

But  because  of  the  direct  current  balance  (1)  this  reduces  to 

L   =  PSC.  (6) 

and  the  bridge  is  balanced  when  S  has  the  value  required  by 
(6)  in  addition  to  the  requirement  of  (1). 

PRACTICAL  APPLICATIONS  OF  MAXWELL'S  METHOD 

179.  Case  A.     Self  Inductance  Varied.     Calibration   of  a 
Variable  Standard  of  Self  Inductance. — The  double  balance 
required  in  the  previous  experiment  makes  the  method  long 
and  tedious.     But  if  Q  is  a  variable  self  inductance,  some  value 
of  L  can  be  found  for  each  steady  current  balance.     The 
method  is  therefore  very  useful  in  the  calibration  of  a  variable 
self  inductance. 

The  operation  of  the  calibration  is  simple.  A  direct  current 
balance  is  obtained  with  values  of  P,  S  and  C  which  will  give  a 
product  equal  to  the  value  of  L  it  is  desired  to  calibrate.  Then 
turning  to  alternating  currents,  the  inductance  is  varied  until  a 
balance  is  obtained.  If  L'  is  the  reading  at  this  point,  the 
correction  to  be  applied  is 

,    c  =  PSC  -  L'. 

A  calibration  curve  should  be  plotted  between  L'  as  abscissae 
and  the  corresponding  values  of  c  as  ordinates. 

180.  Case  B.    Resistance  Varied.    Inductance  of  a  Single 
CdiL — In  case  the  self  inductance  to  be  measured  is  not  varia- 
ble, but,  say,  is  a  coil  having  a  fixed  and  constant  valu  e  for  L, 
the  balance  can  readily  be  obtained  as  follows :    Let  R  and  S, 
Fig.  82,  be  two  similar  decade  resistance  boxes  so  that  the 
resistances  of  these  two  arms  of  the  bridge  can  easily  be  kept 
equal  to  each  other.     Start  with  some  convenient  value,  such 
as  R  =  S  =  1000  ohms. 


MEASUREMENT  OF  INDUCTANCE  209 

In  order  to  obtain  the  direct  current  balance  it  will  now 
be  necessary  to  make  P  =  Q.  Usually  Q  is  not  very  large,  and 
if  P  is  a  resistance  box  the  balance  can  be  obtained  only  to  the 
nearest  ohm.  The  final  balance  can  be  made  by  adjusting  a 
low  resistance  rheostat  in  series  with  the  coil  and  forming  a 
part  of  the  arm  Q.  The  amount  of  resistance  used  from  this 
rheostat,  as  well  as  the  resistance  of  the  coil  itself,  need  not  be 
known. 

The  battery  and  galvanometer  are  now  replaced  by  an  alter- 
nating E.M.F.  and  a  current  detector.  By  varying  R  and  S 
together,  thus  keeping  them  equal,  the  alternating  current  bal- 
ance is  readily  found  and  the  value  of  S  giving  this  balance  is 
the  one  required  in  Eq.  6.  If  R  and  S  are  two  similar  decade 
boxes,  as  specified  above,  they  can  be  kept  equal  and  the 
balance  found  as  easily  as  though  a  single  resistance  was 
varied.  If  several  independent  determinations  of  L  are 
required,  a  small  resistance  may  be  added  to  Q  each  time  and 
the  measurements  repeated. 

If  it  is  desired  to  measure  L  when  the  coil  is  carrying  a 
certain  constant  current,  P  and  R  can  be  set  equal  and  the 
balance  obtained  by  adjusting  S  and  Q  in  the  same  manner  as 
before.  By  maintaining  a  constant  alternating  current  volt- 
age at  AD,  the  current  through  Q  will  remain  constant  while 
P  and  R  are  being  adjusted  to  give  the  alternating  current 
balance.  This  arrangement  is  especially  desirable  when  the 
coil  has  an  iron  core,  or  whenever  the  value  of  L  depends  upon 
the  current  through  the  coil. 

181.  Case  C.  Capacity  Varied. — If  the  capacity  of  the 
condenser  C  can  be  changed  by  several  small  steps,  it  is  often, 
convenient  to  obtain  the  alternating  current  balance  by  vary- 
ing the  amount  of  capacity  in  parallel  with  the  resistance  R. 
Changing  the  capacity  of  C  will  have  no  effect  on  the  resistances 
of  the  bridge,  and  therefore  if  once  in  balance  it  will  not  be 
disturbed  by  making  the  alternating  current  balance. 

If  the  steps  are  too  large  to  give  an  exact  balance  with 
varying  currents,  the  two  values  nearest  the  balance  can  be 

14 


210  ELECTRICAL  MEASUREMENTS 

tried  and  the  deflections  noted.  Then  the  value  of  the  capacity 
which  would  give  the  exact  balance  can  be  determined  by 
interpolation.  This  value  of  C  is  to  be  used  in  eq.  6  to  com- 
pute the  value  of  L. 

182.  Case  D.  Effect  of  Capacity  Varied.  —  Sometimes  it  is 
necessary  to  measure  the  self  inductance  of  a  coil  with  a  con- 
denser of  fixed  capacity.  In  this  case  the  effect  of  the  capa- 
city can  be  varied  by  putting  the  condenser  in  parallel  with 
only  a  part  of  R.  Let  R  consist  of  two  resistance  boxes, 
V  and  TF,  so  arranged  that  while  the  resistance  in  either  may 
be  varied,  yet  the  total  resistance  in  both  shall  always  be  R. 
Let  the  condenser  be  placed  in  parallel  with  V  only.  The 
potential  to  which  the  condenser  is  now  charged  is  only  V/R 
of  what  it  was  before,  and  therefore  the  quantity  it  will  receive 
is  the  same  fraction  of  its  previous  charge.  Furthermore,  the 
effect  of  the  condenser  in  reducing  the  initial  fall  of  potential 
in  this  arm  of  the  bridge  will  be  only  over  the  part  V  instead 
of  over  the  whole  of  R  as  before.  Hence  again,  the  effect  of  the 
condenser  is  reduced  to  V/R  of  its  former  amount. 

The  effect,  therefore,  of  a  capacity  Cf  when  in  parallel  with 
F  only  is  the  same  as  that  of  a  capacity 

V2 


when  in  parallel  with  all  of  R.    Putting  this  value  of  C  into  eq. 
6  gives, 

F2 
' 


as  the  modified  form  of  the  equation. 

183.  Anderson's  Method  for  Comparing  a  Self  Inductance 
with  a  Capacity.  —  In  Anderson's  modification  of  Maxwell's 
method,  the  effect  of  the  condenser  is  varied  without  disturbing 
the  direct  current  balance.  The  setup  differs  from  Maxwell's 
method  by  having  a  resistance,  r,  in  series  with  the  condenser, 
and  then  connecting  the  galvanometer  at  F,  Fig.  83,  instead  of 


INDUCTANCE  OF  MEASUREMENT 


211 


at  E.  This  makes  no  difference  with  the  steady  current 
balance,  but  with  varying  currents  the  effect  of  the  condenser 
can  be  varied  by  changing  the  value  of  r.  The  alternating 
current  generator  with  its  commutator  for  the  galvanom- 
eter circuit  can  be  used  to  advantage. 

When  r  has  been  adjusted  to  give  no  current  through  the 


FIG.  83. — Comparison  of  capacity  and  self  inductance. 

galvanometer  branch,  the  potentials  of  B  and  F  must  always 
remain  equal.     Then 


P(n  +  0  +  n  =  «»,  +  L 


(i) 


where  the  currents  are  as  shown  in  the  figure.     Similarly  for 
the  other  branches 


Si, 


and 


(2) 
(3) 


212 


ELECTRICAL  MEASUREMENTS 


Substituting  in  (1)  the  values  of  i\  and  iz  from  (2)  and  (3) 
gives 

P 

R 


or, 
since 


Making  use  of  the  fact  that  PS  =  QR  from  the  steady  cur- 
rent balance,  (4)  becomes, 

(5) 


which  reduces  to  Maxwell's  value  when  r  =  0. 


MEASUREMENT  OF  MUTUAL  INDUCTANCE 

184.  Direct  Comparison  of  Two   Mutual  Inductances. — 
Mutual  inductances  are  readily  measured  by  comparing  the 

E.M.F's.  induced  in  their  secondaries. 
If  the  current  through  the  primary 
coils  is  direct  the  induced  E.M.F. 
will  appear  at  the  make  and  bieak  of 
the  circuit;  while  if  alternating  current 
is  used  there  will  be  an  alternating 
E.M.F.  in  the  secondary. 

In  the  direct  comparison  of  two  mu- 
tual inductances  the  two  primaries  are 
joined  in  series  and  connected  to  the 

source  of  current,  preferably  a  low-voltage  alternating  circuit. 
The  two  secondaries  are  also  joined  in  series,  with  the  two  in- 
duced E.M.F's.  opposed  to  each  other.  This  method  requires 
a  variable  standard  of  mutual  inductance  and  a  telephone,  or 
a  galvanometer  and  a  rotating  commutator,  to  indicate  zero 
current  in  the  secondary  circuit.  By  turning  the  movable 


FIG.  84. — Comparison 
of  two  equal  mutual  in- 
ductances. 


MEASUREMENT  OF  INDUCTANCE 


213 


coil  of  the  standard  until  the  E.M.F.  induced  in  it  is  just  equal 
and  opposite  to  that  induced  in  the  secondary  of  the  other  pair 
of  coils,  as  shown  by  zero  deflection  of  the  galvanometer,  the 
value  of  the  mutual  inductance  can  be  read  directly  from  the 
standard.  This  is  evident  from  the  following  considerations. 
Writing  KirchhofFs  law  for  the  complete  secondary  circuit 
when  not  in  balance  gives,  at  any  instant,  t, 


iir/  T  ,          „. 

M    ~  Rl  ~  L   ~  Ot  ~ 


di       T,r,,dl 

~  M    ~ 


where  G  and  Lg  refer  to  the  galvanometer. 

For  a  balance  i  =  0,  and  is  constant. 
Therefore, 

M'  =  M" 

It  thus  appears  that  only  mutual  inductances  of  the  same 
value  as  the  standard,  that  is,  up  to  10  millihenrys,  can  be 
measured  by  this  method.  However  the  range  can  be  ex- 
tended by  adding  to  the  variable  standard  another  mutual 
inductance  of  fixed  value.  This  is  done  by  joining  the 
primaries  in  series  -and  also  the  secon- 
daries, thus  adding  together  the 
E.M.F.'s  induced  in  the  two  secon- 
daries. 

185.  Comparison  of  Two  Unequal 
Mutual  Inductances.  —  Let  M  be  a  pair 
of  coils  whose  mutual  inductance  is 
known  and  M'  another  pair  whose  mu- 
tual inductance  is  desired.  The  pri- 
maries of  the  two  coils  are  joined  in 
series  with  a  battery  and  key,  the  coils 
themselves  being  placed  as  far  apart 

as  possible  and  at  right  angles  to  each  other  so  that  each 
secondary  will  be  influenced  only  by  its  own  primary.  The 
two  secondaries  are  also  joined  in  series  with  two  resistance 
boxes,  and  a  galvanometer  is  connected  across  from  B  to  C. 
This  is  not  a  Wheatstone  bridge  arrangement  for  the  two 


two  mutual  inductances. 


T£ 

du 


214  ELECTRICAL  MEASUREMENTS 

electromotive  forces  in  the  two  secondaries  act  together  and 
the  current  flows  through  the  four  arms  in  series. 

By  properly  adjusting  R  i  and  R  2  the  galvanometer  will  give 
no  deflection  when  K  is  opened  or  closed.  Writing  out  Kirch- 
hoff's  law  for  the  circuit  ABC  gives,  at  the  instant  t, 

M^  -  Ri  -  Lft   -  R,i  -  Gi'  -  Lg~  =  0          (a) 
and  for  BDC 

M'dl  -P'i"          J'di"  -i-  04'  -L    T     &i'          T>    •»          r  n  &>* 

Mdi'~Rl    ~L  ~dt+Gl  +  Lg  ~&t  ~~R*    ~L  ~di~- 

where  Lg  denotes  the  inductance  of  the  galvanometer  circuit. 
Integrating  each  of  these  equations  from  the  time  when  the 
key  is  first  closed  and  I  =  0,  to  the  time  when  the  primary 
current  has  reached  its  steady  value,  I  =  /o,  gives, 

(R  +  Ri)q  =  MIQ,  and  (Rf  +  RJq*  =  M'I0, 


if  qf,  the  integrated  current  through  the  galvanometer,  is  zero. 
Dividing  one  equation  by  the  other  gives 

M  _R+Ri 
Mf 


In  this  method  the  galvanometer  current  may  be  zero,  but 
in  general  it  is  the  sum  of  two  transient  currents  each  of  which 
has  an  effect  upon  the  galvanometer  even  when  following  one 
another  in  a  short  interval  of  time.  This  produces  an  un- 
steadiness of  the  galvanometer  and  renders  an  exact  setting 
difficult,  if  not  impossible.  In  order  that  no  current  should 
pass  through  the  galvanometer  it  is  necessary  that  the  poten- 
tial difference  between  B  and  C  shall  remain  zero  for  each 
instant  while  the  primary  current  is  changing,  and  this 
requires  that  the  self  inductance  of  each  branch  shall  be 
proportional  to  theE.M.F.  induced  in  that  part  of  the  circuit. 

Usually  this  is  not  the  case,  but  it  is  not  difficult  to  add  some 
self  inductance,  L",  in  the  part  of  the  circuit  that  is  deficient 


MEASUREMENT  OF  INDUCTANCE  215 

and  thus  fulfill  this  condition.  The  galvanometer  will  then 
indicate  a  much  closer  balance,  or  it  may  be  replaced  by  a 
telephone  and  an  alternating  current  used  in  the  primaries. 

If  the  apparatus  is  at  hand,  the  telephone  may  well  be 
replaced  by  a  galvanometer  and  rotating  commutator  which 
will  reverse  the  galvanometer  terminals  as  often  as  the  alter- 
nating current  is  reversed.  This  is  more  sensitive  than  the 
telephone  but  it  can  be  used  only  where  the  instantaneous 
galvanometer  current  is  zero  for  a  balance.  In  other  words  this 
arrangement  indicates  zero  current.  It  could  not  be  used  in 
the  first  arrangement  where  a  balance  was  indicated  by  zero 
value  of  the  integrated  current. 

185a.  Comparison  of  a  Large  Mutual  Inductance  with  a 
Small  One. — When  the  two  mutual  inductances  differ  very 
widely  in  value  it  is  not  always  possible  or  convenient  to  deter- 
mine their  ratio  directly  as  in  the  previous  method.  Two 
methods  of  reducing  the  effect  of  the  larger  inductance  are 
available,  and  both  are  here  outlined. 

First. — Refering  to  Fig.  84,  in  the  method  for  comparing  two 
equal  mutual  inductances  the  E.M.F's.  induced  in  the  two 
secondaries  were  made  equal  by  varying  one  of  the  inductances, 
When  this  cannot  be  done,  the  balance  can  still  be  obtained  by 
reducing  the  primary  current  in  the  larger  mutual  inductance 
by  means  of  a  shunt 

Then, 

MI  =  MT, 
and, 

M     r      P 


M '  ~ I   ~  P  +  Q 

where  Q  and  P  are  resistances  of  the  primary  coil  and  its  shunt 
respectively      See  Fig.  86. 

Second. — By  shunting  the  secondary  of  the  stronger  mutual 
inductance,   as  indicated  in  Fig.  86,  the  effect  of  its  larger 


216 


ELECTRICAL  MEASUREMENTS 


E.M.F.  can  be  balanced  against  the  smaller  E.M.F.  of  the 
other  coil. 

Writing  out  Kirchoff 's  law  for  the  galvanometer  circuit  gives. 

Mft  ~  Ri  ~L%-  «•*'  -  L»  I"  -  S(i  +  **>  -  °- 

The  corresponding  equation  for  the  circuit  consisting  of  the 
other  coil  and  its  shunt,  is, 


FIG.  86. — Comparison  of  two  very  unequal  mutual  inductances. 

"'f'-Bv-i'f-w  +  o-o. 

Integrating  these  equations  from  the  time  that  the  primary 
circuit  is  closed  until  the  primary  current  has  reached  its  final 
steady  value,  gives, 

MI  -  (R  +  Ra)   (idt  -0-  S  \  (idt  +    ( i'dt\  =  C  =  0. 
and 

MT  -  R'  (i'dt  -  S  I    I  idt  +    f  i'dt    =  C"  =  0. 


MEASUREMENT  OF  INDUCTANCE  217 

The  constants  of  integration  are  zero  for  the  case  when  the 
final  steady  value  of  the  primary  current  is  zero,  and  hence 
they  must  always  be  zero. 

If  the  galvanometer  deflection  is  zero  it  shows  that  the  total 

quantity,    j  idt,  that   passed  through  the  galvanometer  is 

zero.  Putting  this  value  into  the  equations  above,  they 
become, 

MI  =  SJi'dt 

and 

MT  =  (Rf  +  S)   (i'dt 

Dividing  the  first  by  the  second  gives, 

M       s     r      s       P 


M     R'  +  S  I     R'  +  S  P  +  Q 

where  the  relation  between  the  final  steady  values  of  the 
currents  /  and  /'  in  the  primary  circuit  is  given  by  the  usual 
law  of  shunts. 

This  is  the  relation  between  the  two  mutual  inductances 
when  both  shunts  are  used.  If  either  one  is  omitted  ( =  °° ) 
the  corresponding  factor  becomes  unity. 

186.  Measurement  of  a  Mutual  Inductance  in  Terms  of  a 
known  Capacity.  Carey  Foster's  Method. — In  this  arrange- 
ment, shown  in  Fig.  87,  one  pair  of  coils  is  replaced  by  a  con- 
denser, and  the  transient  current  from  the  secondary  coil  S 
is  balanced  against  the  current  that  is  charging  the  condenser 
C. 

If  the  condenser  were  removed  there  would  be  a  current  from 
S  through  R'  and  the  galvanometer  each  time  the  key  in  the 
primary  circuit  is  closed,  and  a  current  in  the  opposite  direction 
when  the  key  is  opened.  With  the  condenser  in  place  this 
current  arrives  at  C  just  in  time  to  charge  the  condenser,  or  to 
help  charge  it.  The  final  charge  in  the  condenser  is 

Q=  C  X  V  =  CRI  (1) 


218 


ELECTRICAL  MEASUREMENTS 


where  /  is  the  final  steady  current  through  R.  The  amount 
of  this  charge  is  the  same  whether  there  is  a  current  through 
the  galvanometer  or  not.  By  adjusting  the  resistance  in  R', 
the  current  through  it  can  be  made  too  small,  too  large,  or 
just  sufficient  to  supply  the  charge  in  the  condenser,  the  rest 
of  the  charge  coming  directly  through  the  galvanometer. 
When  there  is  no  deflection  of  the  galvanometer  the  adjust- 
ment is  complete,  and  the  arrangement  is  said  to  be  balanced. 
Writing  out  Kirchhoff's  law  for  the  circuit  through  SR'G, 
gives, 


Integrating  this  between  the  limits  of  time  t'  just  before  K  is 
closed,  to  t",  when  the  primary  current  has  reached  its  steady 
value  /,  gives, 


Fia.  87. — Comparison  of  a  capacity  and  a  mutual  inductance. 

MI  -  (S  +  R')Qf  +  Gq  =  0,  (3) 

where  Q'  is  the  total  quantity  passing  through  R',  and  q  is  the 
quantity  through  the  galvanometer.  Then,  Q'  +  q  =  Q,  the 
total  charge  in  the  condenser.  But  if  the  galvanometer  deflec- 
tion is  zero,  then  q  =  0,  and 

MI  =  (S  +  R')Q  =  (S  +  Rf)  CRI 
from  (1).     Hence 

M  =  (S  +  R')  CR 


MEASUREMENT  OF  INDUCTANCE  219 

It  is  to  be  noted  that  the  galvanometer  current  is  not  re- 
quired to  be  zero  for  a  balance,  and  in  fact  it  usually  is  not 
zero  for  each  instant  from  t'  to  t" '.  Zero  deflection  merely 
indicates  that  the  algebraic  sum  of  the  quantities  passing 
through  the  galvanometer  is  zero.  Nevertheless,  the  more 
nearly  the  galvanometer  currents  are  to  zero  at  each  instant 
the  more  steady  will  be  the  zero  deflection  at  the  balance. 


CHAPTER  XV 
ALTERNATING  CURRENTS 

187.  An  Alternating  Current  is  the  same  as  any  other  electric 
current,  except  that  it  flows  in  one  direction  for  only  a  very 
short  time;  it  then  reverses  and  flows  in  the  other  direction  for 
an  equally  short  time.     In  ordinary  lighting  circuits  there  are 
from  100  to  300  such  reversals  each  second.     In  some  other 
cases  there  may  be  many  millions  of  reversals  each  second. 
While  the  current  is  flowing  in  one  direction  it  is  the  same  as 
any  other  current  of  the  same  number  of  amperes.     The  only 
peculiarity  of  an  alternating  current  is  that  it  is  continually 
being  made  to  change.     And  just  as  a  material  body  cannot 
change  its  velocity  from  one  direction  to  the  opposite  without 
first  slowing  down  to  zero  and  then  starting  up  in  the  other 
direction,  so  the   current  cannot  instantly  change  from  its 
full  value  in  one  direction  to  the  full  value  in  the  other,  but  it 
requires  some  time  to  die  down  to  zero  and  then  to  build  up  in 
the  opposite  direction.     It  does  not  have  time  to  build  up  very 
far  before  it  must  begin  to  decrease  again,  so  there  is  never 
a  time  when  the  current  is  not  changing  in  amount.     In  fact, 
the  value  of  the  current  as  it  changes  from  one  direction  to  the 
other  and  back  again  goes  through  the  same  variations  as  the 
velocity  of  a  pendulum  bob  when  swinging  to  and  fro. 

188.  Tracing  Alternating  Current  and  E.M.F.  Curves. — In 
Chapter  VII  there  were  given  some  methods  for  measuring  the 
current  flowing  through  a  circuit.     The  same  arrangements  can 
be  used  to  measure  the  value  of  an  alternating  current.     Since 
the  balance  point  would  vary  rapidly  up  and  down  the  slide 
wire  of  the  potentiometer  it  will  be  necessary  to  add  some 
mechanical  device  that  will  close  the  galvanometer  circuit 

220 


ALTERNATING  CURRENTS 


221 


for  only  an  instant  at  the  particular  time  when  the  current  has 
the  value  that  it  is  desired  to  measure.  As  the  current  will 
have  the  same  value  sixty  (say)  times  in  each  second,  the 
galvanometer  circuit  can  be  closed  sixty  times  each  second,  and 
this  is  often  enough  to  produce  a  steady  deflection  of  the 
galvanometer  when  the  potentiometer  is  not  balanced.  Thus 
the  actual  setting  is  made  as  easily  as  when  the  current  is 
steady. 

Let  AD  represent  the  slide  wire  potentiometer,  S  the  resist- 
ance through  which  the  alternating  current  is  flowing,  and  M 
the  instantaneous  contact  maker  which  closes  the  galvanometer 


FIG.  88. — Potentiometer  for  tracing  a.c.  curves. 

circuit  for  a  very  short  time  once  in  each  cycle.  Let  i  denote 
the  value  of  the  alternating  current  at  the  instant  the  gal- 
vanometer circuit  is  closed  by  M.  The  fall  of  potential  over  S 
is  then  Si,  and  if  C  is  moved  so  that  the  fall  of  potential  over 
AC  is  the  same  as  Si  there  will  be  no  deflection  of  the  gal- 
vanometer. Therefore  the  distance  AC  is  proportional  to  the 
current  i. 

Now  let  the  contact  maker  M  be  turned  a  few  degrees  so  as 
to  close  the  circuit  just  a  little  later  than  before.  The  value 
of  the  current  at  this  point  will  be  somewhat  different,  and  its 
value  can  be  found  by  moving  C  along  till  a  balance  is  again 
obtained.  In  this  way  the  values  of  the  current  for  the  com- 


222  ELECTRICAL  MEASUREMENTS 

plete  cycle  can  be  determined,  and  a  curve  plotted  showing 
how  the  current  varies  with  the  time. 

When  the  current  reverses  and  the  fall  of  potential  over  S  is 
in  the  other  direction,  C  must  be  moved  to  the  other  side  of  A 
to  find  a  balance.  Therefore  AD'  is  merely  a  second  slide  wire 
potentiometer  on  which  all  negative  values  of  the  current  can 
be  measured.  In  this  manner  the  values  of  the  current  at 
various  instants  corresponding  to  the  different  settings  of  M 
can  be  measured.  These  should  extend  far  enough  to  com- 
plete at  least  one  cycle,  that  is,  until  the  readings  begin  to 
repeat  themselves.  If  the  dynamo  that  is  generating  the  cur- 
rent has  two  poles  there  will  be  one  cycle  for  each  revolution 
of  the  armature.  If  there  are  more  poles,  there  will  be  as 
many  cycles  in  each  revolution  as  there  are  pairs  of  poles. 

It  is  absolutely  necessary  that  the  instantaneous  contact 
maker,  M ,  closes  the  galvanometer  circuit  at  precisely  the  same 
point  in  each  cycle.  The  contact  maker  is  therefore  placed  on 
the  dynamo  shaft,  but  there  is  no  electrical  connection  between 
the  two;  but  this  insures  that  the  contact  maker  will  keep  rigid 
step  with  the  current. 

The  curve  for  the  alternating  E.M.F.  can  be  traced  in  the 
same  manner.  Usually  the  full  E.M.F.  is  too  large  to  be 
measured  directly  by  the  potentiometer,  but  by  the  fall  of 
potential  method,  as  used  in  the  method  for  calibrating  a  high 
reading  voltmeter,  any  small  portion  of  this  E.M.F.  can  be 
obtained.  It  is  then  only  a  matter  of  increasing  the  scale  to 
get  the  curve  of  the  full  E.M.F.  When  the  current  and  E.M.F. 
curves  are  both  drawn  on  the  same  sheet  the  phase  relation 
between  them  is  clearly  shown. 

After  tracing  the  curves  for  a  current  flowing  through  a 
non-inductive  resistance  it  will  be  interesting  to  do  the  same 
for  a  coil  having  considerable  inductance.  This  should  show 
the  effect  of  inductance  in  making  the  current  "lag  behind  the 
E.M.F. "  If  desired  the  experiment  can  be  further  varied  by 
connecting  a  condenser  in  the  circuit  and  finding  the  effect  it 
has  upon  the  current  and  E.M.F.  curves. 


ALTERNATING  CURRENTS  223 

189.  Measurement  of  an  Alternating  Current.  —  The  usual 
alternating  current  follows  closely  the  sine  law,  and  its  value 
at  any  instant  is  given  by  the  equation, 

i  —  I  sin  a)t 

where  /  and  o>  are  constants.1 

Instruments,  however,  are  seldom  made  to  give  the  instan- 
taneous value  of  the  cuirent,  but  they  record  some  kind  of  an 
average  value.  Evidently  the  arithmetical  average  of  the 
values  of  a  current  which  is  negative  as  much  as  it  is  positive, 
would  be  zero  and  therefore  such  an  instrument  as  a  galvanom- 
eter, or  an  ordinary  ammeter,  would  be  useless  for  the  meas- 
urement of  an  alternating  current.  But  an  electrodynamom- 
eter,  or  the  Kelvin  balance,  measures  the  current  equally 
well  whichever  way  the  current  flows  through  it. 

189A.  Instantaneous  Values  of  the  Current.  —  When  an 
alternating  current  flows  through  a  circuit  there  are  induced 
other  E.M.F's.  besides  that  impressed  by  the  dynamo;  and 
these  extra  E.M.F's.  often  have  considerable  influence  in  deter- 
mining what  the  resulting  current  shall  be.  As  we  have  seen, 

Article  172,  because  of  the  fact  that  the  current  is  changing 

di 
at  the  rate  of  -r  amperes  per  second,  there  will  be  induced  in  the 

dt 

di 

circuit  an  E.M.F.  of  —L  -7:  volts.     In  order  to  maintain  the 

at 

current  i,  the  dynamo  must  furnish  not  only  the  E.M.F.  Ri, 
required  by  Ohm's  law,  but  in  addition  must  supply  an 
E.M.F.  sufficient  to  counter  balance,  at  each  instant,  this 
induced  E.M.F.  That  is,  the  dynamo  must  supply  an  E.M.F. 
whose  value  at  any  instant  is 


This  is  the  general  equation  for  any  current  varying  in  any 

1  In  case  the  current  can  not  be  represented  by  a  single  term  it  can 
always  be  expressed  by  a  series  of  such  terms. 


224  ELECTRICAL  MEASUREMENTS 

manner  whatsoever.     Solving  this  equation  for  the  current 
gives, 

rdi 
e~Ldt 


which  shows  that  the  instantaneous  value  of  the  current  is 
given  by  the  usual  form  of  Ohm's  law  —  taking  into  account  all 
of  the  E.M.F's.  in  the  circuit  at  that  instant. 

190.  Definition  of  an  Ampere  of  Alternating  Current.  —  It 
is  evident  that  an  alternating  current  has  no  steady  value,  and 
to  speak  of  one  ampere  of  alternating  current  is  meaningless 
without  some  definite  convention  or  definition.     An  alternat- 
ing current  will  heat  a  wire,  or  the  filament  of  an  electric 
lamp,  through  which  it  passes,  the  heat  produced  each  second 
being  given  by  the  formula,  Ri2.     Although  this  value  fluctu- 
ates widely  the  light  from  the  filament  appears  continuous. 
By  convention  then,  one  ampere  of  alternating  current  is  that 
amount  which  will  bring  a  lamp  to  the  same  brightness  as  one 
ampere  of  steady  direct  current. 

In  a  hot  wire  ammeter  the  current  is  measured  by  the  in- 
crease in  length  of  a  fine  wire  due  to  the  heat  produced  in  it 
by  the  passage  of  the  alternating  current. 

If  an  electrodynamometer  has  been  calibrated  for  direct 
current  it  may  be  used  to  measure  alternating  current,  and 
that  amount  of  current  which  will  give  the  same  deflection  as 
one  ampere  of  direct  current  is  called  one  ampere  of  alternating 
current. 

191.  Average    Value    of    a   Sine    Current.  —  To   find   the 
average  ordinate  of  the  sine  curve  it  is  only  necessary  to 
determine  the  area  included  between  the  curve  and  the  axis 
of  abscissae,  and  divide  this  by  the  length  of  the  base.     Then 

area  dx 

Average  ordinate  = 


(    y 
=  1     ~ 
Jo 


But  for  a  sine  curve, 

y  =  a  sin  x 


ALTERNATING  CURRENTS  225 


Hence, 

Pa  C"  .                       a  I                    2a 
Avprn.crp.  nrnirm.T.p    =   —   I     sin  cf.  n.'r.  =     —  —   I       Pins  T.   =   

'Jo 


Average  ordinate  =  -  |    sin  x  dx  =   —  -       cos  x  = 

7T    I  7T 


.0 

=  0.6369  of  maximum  ordinate. 

192.  Mean  Square  Value  of  a  Sine  Current. — Inasmuch  as 
the  heating  and  power  effects  of  an  electric  current,  as  well  as 
the  dynamometer  readings,  depend  upon  the  square  of  the 
values  of  the  current,  it  is  more  useful  to  know  the  average 
value  of  the  square  of  the  current  than  merely  its  average  value. 
This  can  be  found  in  the  same  way  as  before.  Since  the 
squares  of  negative  quantities  are  positive  we  are  not  limited  to 
half  a  period,  but  can  extend  the  integration  over  the  whole 
period. 

J*y2    fly.  a2        C2* 
=   2~  I     sin2  xdx 
o  ^J0 

But,  2  smzx  =  1  —  cos  2x,  and 


Hence,  mean  square  of  y  =  -^,  and  the  square   root  of   the 

A 

mean   square   value  of    the  current  is  0.707  of  maximum 
value. 

Alternating  current  ammeters  and  voltmeters  are  cal- 
ibrated to  read  this  value  of  the  current  because  the  power 
expended  in  a  circuit  depends  upon  the  square  of  the  current. 
The  heat  produced  in  a  hot  wire  ammeter  is  evidently  pro- 
portional to  the  average  square  of  the  current,  and  the  deflection 
of  an  electrodynamometer  (Art.  78)  likewise  depends  upon  the 
average  square  of  the  current.  Therefore  when  such  instru- 
ments are  calibrated  to  measure  direct  currents,  if  they  are  used 

15 


226  ELECTRICAL  MEASUREMENTS 

to  measure  alternating  current  they  will  indicate  the  square  root 
of  the  instantaneous  values  of  the  current.  This  value  is  often 
called  the  "mean  square"  value  for  short.  Sometimes  it  is 
called  the  "  effective  value." 

193.  To  Find  what  E.M.F.  is  Required  to  Maintain  a 
given  Current.  —  Let  an  alternating  current  flowing  in  a 
circuit  containing  both  resistance  and  self  inductance,  be 
given  by  the  equation 

i  —  I  sin  ut  (1) 

It  is  required  to  find  a  similar  expression  for  the  E.M.F.  which 
will  maintain  this  current. 
The  general  equation  is, 

&1 

e  =  Ri  +  L  -jiJ 

where  Ri  is  the  E.M.F.  required  to  maintain  the  current 
through  the  ohmic  resistance,  and  L  -r,  is  the  E.M.F.  to  bal- 

ance the  induced  E.M.F. 

Substituting  the  above  value  of  the  current  gives, 

e  =  RI  sin  co£  +  Leo/  cos  otf,  (2) 

=  RI  sin  cot  +  LcoJ  sin  (co£  -f  90°) 
=  k  sin  (cot  +  a)  (3) 

where  k  and  a  are  new  constants. 

The  values  of  k  and  a  can  be  determined  in  terms  or  R,  L 
and  co  by  expanding  sin  (co£  -f-  a)  as  follows: 

k  sin  (co£  +  a)  =  k  cos  a  sin  ut  +  k  sin  a  cos  wt       (4) 
Comparing  this  with  (2)  it  is  seen  that 

k  cos  a  =  RI,     and     k  sin  a  =  Leo/, 
from  which 


k  =  I  V#2  +  L2w2      and     a  =  tan-1-  (5) 

From  Eq.  3  it  is  evident  that  the  maximum  value  that  e  can 
ever  have  is  k;  or  writing  E  for  this  maximum  value  of  e, 


E  =  IVR2  +  LW.  (6) 

1  See  Article  173. 


ALTERNATING  CURRENTS 


227 


194.  Graphical  Solutions.  —  The  term  RI  sin  ut  in  (2)  may  be 
represented  by  a  line  OA,  Fig.  89.  If  this  line  is  considered 
as  rotating  counter  clockwise  with  the  angular  velocity  co,  its 
vertical  projection  at  any  instant,  t,  gives  a  length  RI  sin  coZ. 

In  the  same  way  the  term  Loo/  cos  wt  may  be  represented  by 
a  line  OB,  which  must  be  drawn  90°  ahead  of  OA  since  cos  at  = 
sin  (ut  +  90°).  The  value  of  e  at  any  instant  will  be  given  by 
the  sum  of  the  vertical  projections  of  these  two  lines  or  what  is 
the  same  thing,  by  the  vertical  projection  of  OC,  which  is  the 
geometrical  sum  of  OA  and  OB. 

From  the  figure  it  is  evident  that  the  length  of  OC  is 


Lw 
and  it  is  ahead  of  OA  by  the  angle  a  =  tan    ]  -p-« 

Problem.  —  Draw  the  curve  representing  the  current  as  given  hi  (1) 
for  at  least  two  cycles. 

On  the  same  axis  draw  the  two  components  of  the  E.M.F.  as  given 
by  (2)  and  by  addition  obtain  the  curve  for  e. 

The  following  constants  may  be  used,  R  =  2  ohms.     7  =  1  ampere, 
co  =  400,  L  =  0.01  henry. 


Lul 


FIG.  89. — Addition  of  two 
harmonic  quantities. 


FIG.  90.— E.M.F.  triangle. 


Fig.  89  is  called  the  parallelogram  of  E.M.F's.  Since  OB  = 
AC,  the  triangle  of  E.M.F's.,  Fig.  90,  shows  the  same  relation- 
ships. The  side  OA  represents  the  E.M.F.  required  to  keep  the 
current  flowing  through  the  resistance,  R.  This  is  called  the 
effective  component  of  the  E.M.F.  The  side  AC  represents 
the  E.M.F.  required  to  just  balance  and  oppose  the  induced 
E.M.F.  due  to  the  self  inductance  of  the  circuit.  This  is 
called  the  inductive  E.M.F.  sometimes  wattless  E.M.F.  because 


228  ELECTRICAL  MEASUREMENTS 

no  energy  is  required  to  maintain  it.  The  geometrical  sum  of 
these  two  gives  the  total  E.M.F.  that  must  be  supplied  to 
maintain  the  current,  and  it  is  called  the  impressed  E.M.F. 

The  angle  of  lag  of  the  current  behind  the  impressed  E.M.F. 

is  shown  by  the  angle  a.     This  may  have  any  value  from  0° 

to  90°  depending  upon  the  amount  of  inductance  in  the  circuit. 

The  Impedance  Triangle.  —  If  each  side  of  Fig.  90  is  divided 

by  the  current  7,  the  result  will  be  the  triangle  of  resistances 

and  impedance  as  shown  in  Fig.  91. 

The  side  R  denotes  the  resistance,  but 
this  is  not  necessarily  the  same  as  the 
direct-current  resistance.  It  is  the  fac- 

LtD 

tor  which  multiplied  by  the  square  of 

the  current  will  give  the  amount  of  heat 

produced  in  the  circuit. 
'IG*  9trkn^ePed  The  other  side  of  the  triangle,  Leo,  is 

called  the  reactance.  It  is  also  meas- 
ured in  ohms,  but  there  is  no  loss  of  energy  or  production  of 
heat  because  of  it.  The  hypotenuse  is  called  the  impedance. 

195.  Mean  Square  Values.  —  In  all  these  diagrams  7  denotes 
the  maximum  value  of  the  current  and  when  so  used  the  result 
obtained  for  E  will  be  the  maximum  value  of  the  E.M.F.     If 
the  ammeter  reading  is  used  for  7,  the  same  construction  can 
be  used  and  the  result  will  be  the  voltmeter  reading  value  of  E. 
But  in  this  case  the  diagram  no  longer  has  the  significance 
that  was  given  it  in  Fig.  89,  and  it  becomes  merely  a  graphical 
construction  to  reach  a  desired  result. 

196.  Measurement  of  Impedance  by  Ammeter  and  Voltme- 
ter. —  From  what  has  just  been  said  it  will  be  seen  that  the 
impedance  of  a  circuit  is  merely  the  ratio  of  the  impressed 
E.M.F.  to  the  resulting  current.     To  measure  it,  therefore,  it  is 
only  necessary  to  measure  the  voltage  and  current  in  precisely 
the  same  way  as  the  direct-current  resistance  is  measured  by  an 
ammeter  and  a  voltmeter.     The  ratio  gives  the  impedance  or, 

E 


=  Impedance  =  VR2  +  L2«2 


ALTERNATING  CURRENTS  229 

197.  Self  Inductance  by  the  Impedance  Method. — If  the 

values  of  R  and  co  are  known,  this  method  gives  a  means  for 
computing  the  self  inductance,  L,  of  the  circuit.  If  the  current 
passes  through  n  cycles  per  second,  then  co  =  2irn. 

The  direct-current  value  may  be  used  for  R  if  the  wire  is  not 
too  large,  and  if  there  are  no  closed  circuits,  or  masses  of  metal 
or  iron  in  the  vicinity. 

In  case  heat  is  produced,  or  energy  is  otherwise  expended, 
outside  of  R  it  will  be  necessary  to  determine  one  other 
quantity  before  the  impedence  triangle  can  be  drawn.  This 
may  take  the  form,  either  of  finding  the  angle  of  lag  of  the 
current  behind  the  impressed  E.M.F.,  or  of  determining  the 
equivalent  resistance  of  the  circuit — that  is,  the  non-induc- 
tive resistance  in  which  the  same  amount  of  energy  would  be 
expended. 

198.  Impedance  and  Angle  of  Lag  by  the  Three  Voltmeter 
Method. — In  this  method  a  non-inductive  resistance,  capable 


FIG.  92. — Three  voltmeter  method. 

of  carrying  the  current  is  placed  in  series  with  the  impedance 
to  be  measured.  Three  voltmeter  readings  are  taken,  as  nearly 
simultaneously  as  possible,  to  measure  the  voltage  across  each 
the  resistance  and  the  impedance  and  across  both  together. 
It  is  best  to  use  three  voltmeters  as  shown  in  Fig.  92,  but  if 
only  one  is  available  it  may  be  transferred  quickly  from  one 
position  to  another.  If  the  voltmeter  shunts  an  appreciable 
current  from  the  main  circuit,  two  equivalent  resistances 
should  occupy  the  places  of  the  missing  voltmeters. 

The  voltmeter  is  readily  transfered  to  the  various  positions 
by  means  of  two  double  throw  switches,  S  and  T,  as  shown  in 
Fig.  93,  where  Vm  denotes  the  voltmeter,  and  U  and  W  are 


230 


ELECTRICAL  MEASUREMENTS 


resistances,  each  equal  to  to  the  resistance  of  the  voltmeter. 
With  both  switches  thrown  to  the  right  the  voltmeter  is  placed 
across  AC,  and  measures  the  total  voltage  over  both  the  im- 
pedance and  the  non-inductive  resistance.  When  both  switches 
are  thrown  to  the  left  the  voltmeter  is  across  BC,  and  measures 
the  voltage  over  the  impedance  alone.  When  the  switch  S  is 
thrown  to  the  left  and  the  switch  T  is  to  the  right,  the  volt- 
meter is  across  AB,  and  measures  the  voltage  over  the  resist- 
ance R,  only.  The  resistances  V  and  W  are  simultaneoulsy 
transferred  to  the  positions  not  occupied  by  the  voltmeter. 
Referring  to  Fig.  90  it  will  be  seen  that  Ez,  the  reading  of  the 


OOQQQOQQOL 


FIG.  93. — Switches  for  putting  the  voltmeter  in  three  places. 

voltmeter  across  BC,  is  the  E.M.F.  impressed  upon  the  coil, 
and  it  would  be  the  hypotenuse  of  the  corresponding  E.M.F. 
triangle  if  the  other  elements  were  known.  In  the  same  way 
Es  is  the  hypotenuse  of  the  E.M.F.  triangle  for  the  entire  circuit 
AC,  while  for  the  part  AB  in  which  there  is  no  inductance  the 
hypotenuse  coincides  with  the  base  of  the  triangle,  and  is 
measured  by  E\. 

Combining  these  three  E.M.F's.  gives  the  triangle  ABC,  Fig. 
94.  Extending  the  side  AB  until  it  meets  the  perpendicular 
from  C  gives  the  complete  E.M.F.  triangle  for  the  entire  circuit 
AC.  The  angle  DBG  gives  the  lag  of  the  current  behind  Ez, 
and  therefore  BDC  is  the  E.M.F.  triangle  for  the  coil. 


ALTERNATING  CURRENTS 


231 


If  the  current  is  also  known,  either  by  direct  measurement 
or  by  computation  if  R  is  known,  the  impedance  of  the  coil  is 
given  by  the  relation 

Impedance  =  -j-2« 


Rl 


El  B  D 

FIG.  94. — Determination  of  the  angle  of  lag. 

199.  Determination  of  Equivalent  Resistance. — Knowing 
the  value  of  one  side  of  the  impedance  triangle  and  the  angle 
a,  the  other  sides  are  readily  constructed.     The  base  of  this 
triangle,  R,  gives  the  value  of  the  equivalent  resistance  of  the 
coil.     The  effect  of  a  solid  iron  core  is  to  greatly  increase  this 
equivalent  resistance  over  the  ohmic  resistance  of  the  coil  as 
measured   by   direct-current    methods.     The    inductance   is 
likewise  increased.     If  the  core  consists  of  a  bundle  of  fine  iron 
wire  the  increase  of  the  resistance  is  less,  while  the  inductance 
is  greater  than  with  the  solid  core. 

200.  Inductive    Circuits  in    Series. — When  two  inductive 
circuits  are  joined  in  series  the  same  current  must,  of  course, 
flow  through  them  both.    But  in  general  the  E.M.F.  over  one 
will  not  be  in  phase  with  that  over  the  other  and  therefore  the 
total  E.M.F.  required  to  maintain  the  current  will  be  less  than 
the  sum  of  the  two  parts.     This  is  readily  seen  from  the  figure. 
A'B'C'  represents  the  two  inductive  circuits  in  series,  and  the 
diagram  ABC  shows  the  E.M.F.  triangles  for  each  part,  and 
for  the  whole   circuit.     The  triangle  ANB  is  the  E.M.F. 
triangle  for  the  portion  A B}  and  corresponds  to  Fig.  90  for  the 
first  part  of  the  circuit. 


232 


ELECTRICAL  MEASUREMENTS 


Similarly  the  E.M.F.  triangle  for  the  part  BC  is  shown  by 
y  which  is  drawn  in  the  position  shown  because  the  point 
B  in  each  triangle  represents  the  one  point  E'  between  the  two 
parts  of  the  circuit,  and  therefore  it  should  occupy  only  one 
position  on  the  diagram.  The  total  E.M.F.  over  the  entire 
circuit  is  given  then  by  AC,  while  the  E.M.F.  triangle  for  the 
entire  circuit  is  found  by  completing  the  triangle  AKC. 

From  this  construction  it  is  seen  that  the  total  resistance  in 
the  circuit  is  the  sum  of  the  resistances  of  each  part;  and  the 


H     N 


0000000 


A  B'  C' 

FIG.  95. — Two  inductive  circuits  in  series. 

total  inductance  is  the  sum  of  the  separate  inductances.  Of 
course  the  two  parts  are  supposed  to  be  far  enough  apart  to 
avoid  mutual  induction  between  them.  The  angle  of  lag  of 
the  current  behind  the  impressed  E.M.F.  is  intermediate 
between  the  angles  of  lag  in  each  part  of  the  circuit  considered 
separately,  and  is  given  by 

.  Lw 
a  =  tan"1  -^- 


Problems 

1.   Given  two  circuits  in  series,  with  R 
z  =  4  ohms,  Lz  =  0.02  henry,  u  =  400. 


=  18  ohms,  LI  =  0.01  henry, 
Find  the  E.M.F.  necessary  to 


ALTERNATING  CURRENTS  233 

maintain  10  amperes  through  the  circuit;  also  the  E.M.F.  over  each  part. 
Solution. — Draw  to  scale  a  figure  similar  to  Fig.  95.     Then  use  then 
same  scale  to  measure  AC,  AB,  and  BC. 

2.  Given  the  same  circuit  as  above.     What  is  the  value  of  the  current 
when  the  impressed  E.M.F.  is  100  volts? 

Solution. — Draw  the  line  A  C  to  represent  the  value  of  E.     At  A  con- 

L'o> 
struct  the  angle  CAH  =  tan  *  ~T>T~>    as    shown    by  the    dotted    lines, 

Fig.  95.     Extend  AH  to  meet  the  perpendicular  from  C  at  K.     Then 
7  =  AK/R',  where  R'  =  #1  +  Rz. 

Solution. — Second  method.  Assume  a  value  /'  for  the  current  and  find 
the  corresponding  value  E'  for  the  impressed  E.M.F.  as  above.  Then 
E'  :  100  :  :  /':  /. 

3.  When  the  impressed  E.M.F.  is  200  volts,  what  is  the  E.M.F.  over 
each  part  of  the  above  circuit? 

Solution. — Draw  the  triangle  KAC  as  before.  Divide  the  side  AK 
in  the  ratio  of  the  two  resistances,  and  the  side  KC  in  the  ratio  of  the  two 
inductances.  Through  the  points  M  and  N  draw  lines  parallel  to  these 
sides;  their  intersection  locates  the  position  of  B.  The  values  of  AB  and 
BC  can  then  be  measured. 

201.  Inductive  Circuits  in  Parallel. — In  the  case  of  a  divided 
circuit  having  two  or  more  inductances  in  parallel,  it  is  much 
more  difficult  to  calculate  what  part  of  the  current  will  pass 
through  each  branch,  and  graphical  methods  become  more  use- 
ful. The  following  example  for  two  inductive  circuits  in 
parallel  can,  of  course,  be  extended  to  as  many  parallel  circuits 
as  desired. 

Since  each  branch  will  have  the  same  impressed  E.M.F.  the 
hypotenuse  of  each  E.M.F.  triangle  will  be  identical.  Let  this 
be  laid  off  to  scale  as  shown  by  AB,  Fig.  96.  Since  each  tri- 
angle is  right  angled,  it  will  be  inscribed  within  a  semicircle 
drawn  on  AB  as  a  diameter.  From  the  constants  of  the  circuit 
the  angle  of  lag  in  each  branch  can  be  determined.  The  base, 
AN,  of  the  first  triangle  can  then  be  laid  off,  making  this  angle 
with  AB.  The  intersection  of  AN  with  the  semicircle  locates 
the  other  corner  of  this  triangle  at  N,  and  the  line  NB  com- 
pletes the  other  side.  The  value  of  the  current  is /i  =  AN/Ri, 
and  is  laid  off  in  the  direction  AN. 

Similarly,  the  triangle  A  MB  is  laid  out  for  the  other  branch, 


234  ELECTRICAL  MEASUREMENTS 

and  the  value  of  the  current  determined.  The  resultant 
current  is  the  geometrical  sum,  AI,  of  these  two  components, 
and  it  lags  behind  the  impressed  E.M.F.  by  the  angle  HAB. 
The  point  H  where  the  line  A I  cuts  the  semicircle  is  the  right 
angled  corner  of  a  new  triangle,  AHB,  which  represents  the 
resultant  or  equivalent  effect  of  a  single  circuit  which  could 
replace  the  two  parallel  circuits.  The  E.M.F.  AH  is  R'l, 
and  HB  is  Z/co/,  where  Rf  and  L'  denote  the  values  of  the 
resistance  and  the  inductance  of  this  equivalent  circuit.  These 
values  can  be  taken  from  the  figure  as  accurately  as  the  lines 
can  be  measured. 


FIG.  96. — Inductive  circuits  in  parallel. 

Problem.— Given  a  coil  with  Ri  =  22  ohms  and  LI  =  0.03  henry,  in 
parallel  with  a  second  coil  with  #2=8  ohms  and  Lz  =  0.03  heary. 
E  =  100  volts,  and  co  =  400.  Find  the  values  of  the  currents  through 
each  branch  and  in  the  main  circuit;  also  the  equivalent  resistance  and 
the  equivalent  inductance  of  the  two  coils  in  parallel,  and  the  angle  of 
lag  of  each  current  behind  the  impressed  E.M.F. 

202.  Graphical  Solution  for  Circuits  having  Capacity. — In  a 

circuit  having  a  condenser  in  series  with  a  resistance  and  an 
alternating  E.M.F.  the  relation  at  any  instant  is 

e  =  fit  +  ^ ;  (1) 


ALTERNATING  CURRENTS 


235 


where  ^  is  the  difference  of  potential  across  the  condenser  of 
o 

capacity  C  farads,  when  its  charge  is  q  coulombs. 


FIG.  97.     , 

If  the  current  flowing  through  R,  Fig.  97,  and  into  the  con- 
denser is  following  the  sine  law,  its  value  at  any  instant  is 


i  =  I  sin  cot 


Then, 


q  =    I  idt  =    I  7  sin  utdt  = cos  ut 

J  J  « 


(2) 


(3) 


FIG.  98. 
Putting  this  value  in  (1)  gives 


e  =  RI  sin  at  — 


cos 


(4) 


236 


ELECTRICAI   MEASUREMENTS 


The  two  terms  of  this  equation  can  be  represented  by  two 
lines  drawn  at  right  angles,  as  was  done  in  Article  194.  The 
result  will  be  different,  however,  for  after  laying  off  OA,  Fig. 
98,  equal  to  RI,  the  other  side,  OB  must  be  drawn  downward 
in  order  to  represent  the  negative  term  in  (4) .  The  geometrical 
sum  of  these  two  is 


OD  =  E 


1 

C2C02 


FIG.  99. 


and  this  is  behind  OA  by  the  angle 


a 


The  current  is  thus  ahead  of  the  impressed  E.M.F.  by  the 
angle  a. 

The  resultant  current  for  circuits  in  series  and  in  parallel 


ALTERNATING  CURRENTS  237 

can  be  obtained  by  combining  figures  like  Fig.  98  as  was  done 
above  for  inductive  circuits. 

In  case  there  is  both  capacity  and  inductance  in  the  same 
circuit,  the  resultant  E.M.F.  required  to  maintain  the  current 
is  found  by  combining  Fig.  89  with  Fig.  98,  as  shown  in  Fig.  99. 

The  E.M.F's.  Leo/  and  -^r  are  opposed  to  each  other,  and  the 
Ceo 

algebraic  sum  of  these  two  is  combined  with  RI  to  give  the 
resultant  E.M.F.,  OD,  Fig.  99. 

203.  Power  Expended  in  an  Alternating-current  Circuit. — 
The  power  expended  in  a  wire  by  an  alternating  current  flowing 
through  it  is  RP,  where  /  denotes  the  mean  square  value  of  the 
current.  If  this  same  wire  is  wound  into  a  coil  having  a  large 
inductance,  the  amount  of  heat  produced  in  the  wire  will  be  the 
same  as  before  for  the  same  current.  But  it  will  now  require  a 
greater  E.M.F.  to  maintain  the  same  current.  However  there 
is  no  expenditure  of  energy  because  of  the  inductance,  since 
half  of  the  time  the  induced  E.M.F.  opposes  the  current  and 
the  other  half  of  the  time  it  helps  the  current  an  equal  amount. 

The  product  RP  can  be  written  as  RI  X  /,  and  RI  is  the 
E.M.F.  necessary  to  maintain  the  current  through  the  resist- 
ance of  the  circuit.  Therefore  the  power  expended  is 

W  =  RI  X  /  =  El  cos  a 

where  E  cos  a  is  the  component  of  the  impressed  E.M.F.  that  is 
in  phase  with  the  current.  This  factor,  cos  a,  is  called  the 
power  factor  of  the  circuit.  Evidently  the  power  factor  be- 
comes small  for  circuits  having  a  large  amount  of  inductance 
or  of  capacity. 

This  explains  how  a  high  E.M.F.  accompanied  by  a  large 
current  in  an  inductive  circuit  may  yet  involve  a  small  amount 
of  power. 


INDEX 


Absolute  system,  1 
Absorption,  46 

Advantages  of  double  method,  82 
Alternating  current,  average  value, 
224 

curves,  221 

definition  of  ampere,  224 

mean  square  value,  225 
Alternating  currents,  220,  223 

measured  by  electrodyna- 

mometer,  99 
by  hot  wire  ammeter,  92 
Ammeter,  calibration,  116,  117 

hot  wire,  92 

use  of,  10 

for  large  currents,  28 

Weston,  92 
Ampere,  1,  6,  8 

of  alternating  current,  224 

turns,  154 
Angle  of  lag,  227 

by    three-voltmeter    method, 

229 

Available  E.M.F.,  21 
Average  value  of  a  sine  current,  224 

B 

B,  measurement,  177,  184 
Ballistic  galvanometer,  34,  143 

and  condenser,  35 

equation,  149 

with  shunt,  52,  54 
Bar  and  yoke,  156 

double,  175 

Battery,  internal  resistance,  19 
Beginning  of  a  current,  199 
Best  arrangement  for  ammeter  and 
voltmeter,  17 

position  of  balance  on  slide 
wire  bridge,  84 


C.  G.  S.  systems,  1 
Cadmium  standard  cell,  110 


Calibration  of  ammeter,  116,  117 

of  bridge  wire  with  extensions, 
81 

of  electrodynamometer,  99 

of  slide  wire  bridge,  64 

of  a  variable  standard  of  self 
inductance,  208 

of  a  voltmeter,  112,  114,  114 

of  a  wattmeter,  127,  129,  132 
Capacities,  comparison  by  bridge 
method,  136 

by  direct  deflection,  40,  135 

by  Gott's  method,  138  . 

by  method  of  mixtures,  139 
Capacity,  absolute,  149 

in  alternating  current  circuits, 
234 

definition,  32 

unit,  32 

Carey  Foster's  method  for  measur- 
ing resistance,  88 
Charge,  33 
Coercive  force,  180 
Concrete  examples,  7 
Condensers,  33 

laws  of,  13l~ 
Conductivity,  76 

Constant  of  ballistic  galvanometer, 
37,  157,  194 

of    galvanometer    by    known 

mutual  inductance,  195 
Constants  of  galvanometer,  50 
Coulomb,  7,  135 
Coulometer,  95 
Critical  damping,  36 
Crosses,  70,  73 
Current,  alternating,  220,  223 

element,  magnetic  effect,  171 

electric,  3 

galvanometer,  47 

magnetic  effect,  3 

and  magnetic  field,  160,  161 

measurement  of,  92 

by  standard  cell,  115 

starting  and  stopping,  197 

unit,  definition,  4 


239 


240 


INDEX 


Current  by  voltmeter  and  shunt,  27 
Currents,  laws  of,  10 


Damping  of  galvanometer,  36,  147 
Deflection  of  galvanometer,  47 
Definition  of  ampere,  6 

ampere  of  alternating  current, 

224 

coulomb,  7 
farad,  34 
gauss,  2 

gilbert,  193,  197 
henry,  193,  197 
international  units,  8,  9 
joule,  7 
maxwell,  167 
microfarad,  34 
megohm,  50 
oersted,  155 
ohm,  6 

unit  current,  4,  173 
field,  2 

flux  density,  167 
magnetic  induction,  165 
potential  difference,  6 
pole,  2 
quantity,  4 
resistance,  5 
volt,  7 
watt,  7,  8 
Depolarizer,  110 
Difference  of  potential,  5 
Differential  galvanometer,  56,  58 
Direct  reading  bridge,  86 
Double  bar  and  yoke,  175 

method,  slide  wire  bridge,  65 

advantages,  82 
D'Arsonval  galvanometer,  93 
Dying  away  of  a  current,  198 


E 


Effective  E.M.F.,  227 
Efficiency  of  a  cell,  24 

of  electric  lamps,  124 
Electrodynamometer,  98 

as  wattmeter,  119 

for  measurement  of  alternat- 
ing current,  99 
Electrolytes,  resistance  of,  75 
Electromagnetic  induction,  187 

system,  1 


Electromotive  force,  5 
Electrons,  34 
Electrostatic  system,  1 

capacity,  34 
E.M.F.,  induced,  166 

of  cell  by  voltmeter  and  auxil- 
iary battery,  25 

required  to  maintain  an  alter- 
nating current,  226 
E.M.F's.,  by  condenser,  39 

by  potentiometer,  106 
Equivalent  resistance,  231 
Extensions,  slide  wire  bridge,  78 

F 

Fall  of  potential,  5 

in  a  circuit,  12 
Farad,  34 

Faults,  location  of,  69,  71 
Field  intensity,  2 
Figure  of  merit,  47,  48,  49 
Fisher's  method,  72 
Flux  turns,  153 

density,  165,  167,  177 
Force    on    conductor    carrying    a 
current,  165 

G 

Galvanometer,  ballistic,  34 

with  universal  shunt,  54 

D'Arsonval,  93 

description,  47 

differential,  56,  58 

tangent,  94 

use  of,  47 

Gauss,  definition,  2 
Gilbert,  154 

Graphical  solutions,  227 
Grounds,  69,  70 

H 

H,  measurement  of,  177,  185 
Heat  produced  in  electric  circuit,  5 
Henry,  193,  197 
Hysteresis,  180,  181 
loss,  185 


Impedance  by  ammeter  and  volt- 
meter, 228 

by  three-voltmeter  method, 
229 

triangle,  228 


INDEX 


241 


Impressed  E.M.F.,  228 
Induced  E.M.F.,  166 
Induction,  electromagnetic,  187 
Inductive  circuits  in  parallel,  233 
in  series,  231 

E.M.F.,  227 

Insulation  resistance,  44,  45 
Internal  resistance  of  a  battery,  19, 

20,  41 
International  ampere,  8,  9 

farad,  34 

ohm,  8,  9 

volt,  8,  9 

watt,  8 
Ions,  75,  110 

K 

Kelvin  balance,  96 
Kirchhoff's  laws,  100 

second  law,  proof,  103 


Law  of  the  magnetic  circuit,  152 
Laws  of  condensers,  107 

of  electric  currents,  10 

of  resistance,  16,  49 
Length  of  bridge  wire  with  exten- 
sions, 80 
Logarithms,  natural,  201 

M 

Magnetic  circuit,  151 

study  of,  156 
effect  of  a  current,  3 
field,  2 

at  center  of  circle,  173 
and  current,  161,  170 
flux,  152 

measurement,  155,  157,  168 
force,  2 

value  of,  177 
induction,  166,  177 

measurement  of,  177 
pole,  1 
tests,  164 

Magnetomotive  force,  153,  157 
Magnets,  permanent,  186 
Maxwell,  153 

definition,  167 

Maxwell's  method  for  comparing  a 
self  inductance  with  a 
capacity,  206 


Mean  square  values,  228 
Megohm  sensibility,  50 
Microfarad,  34 

M.M.F.  and  field  intensity,  168 
Multiplying  power  of  a  shunt,  51, 

54 

Murray  loop,  71 
Mutual  inductance,  calculation  of. 

193 
compared  with  a  capacity, 

217 

compared  with  a  self  induct- 
ance, 205 

comparison    with    another 
mutual  inductance,   212, 
213,  215 
meaning,  191 
unit,  193 
induction,  laws  of,  187 

N 

Natural  logarithms,  201 
O 

Ohm,  1,  6,  7,  8 
Ohm's  law,  14 
Opens,  69,  70,  74 


Parallelogram  of  E.M.F's.,  227 
Permeability,  2,  178 
Pole,  unit,  definition,  2 
Potential  difference,   unit,   defini- 
tion, 6 

differences,  100 
Potentiometer,  108 
method,  105 

for  calibrating  a  voltmeter, 

114 
for  calibrating  an  ammeter 

117 

resistance  box,  106 
slide  wire,  105 
Power  from  a  cell,  23 
measurement  of,  119 
expended  in  a  rheostat,  123 
in  alternating  current  circuit, 

237 
in  terms  of  a  standard  cell,  125 

Q 

Quantity,  unit,  definition,  4 
electrostatic,  1 


242 


INDEX 


R 

Reluctance,  154 
Residual,  discharge,  142 

magnetism,  180 
Resistance,  5,  14 
laws  of,  17,  63 
measured    by    ammeter    and 

voltmeter,  15,  16 
by  Carey  Foster's  bridge,  87 
by  differential  galvanome- 
ter, 57,  59 

by  potentiometer,  111 
by  slide  wire  bridge,  62,  78 
by  Wheatstone  bridge  box. 

66 

equivalent,  231,  234 
high,  by  voltmeter,  28 
insulation,  44,  45 

by  condenser  method,  41 
of  battery,  19 
of  a  galvanometer,  55,  56 
temperature  coefficient,  89 
unit,  definition,  5 
Resistances  in  series,  17,  63 

in  parallel,  17,  63 
Resistivity,  76 
Resolutions,  London,  7 
Ring  method,  178 
solenoid,  169 

S 

Screw  relation,  3 
Self     inductance     by     impedance 

method,  229 

calibration  of  standard,  208 

compared  with  a  capacity, 

206,  211 

comparison  with  standard, 

meaning,  197 

unit,  197 

of  a  single  coil,  208 
Shunt  box,  test,  54 
common,  51 
universal,  52 
Shunts,  51 

for  ammeters,  28,  93 
Siemens  electrodynamometer,  98 
Slide  wire  bridge,  62 

best  position  for  balance,  84 

calibration,  64 

double  method,  65 

sources  of  error,  86 

with  extensions,  78 


Solenoid,  long,  170 

ring,  169 
Sources  of  error  in  slide  wire  bridge, 

86 
Standard  cells,  109 

use  of,  110 
Starting  and  stopping  a  current, 

197 
Step-by-step  method,  179,  180 


Tangent  galvanometer,  94 
Temperature  coefficient  of  resist- 
ance, 89 

Testing  key.  43,  141 
Time  test  or  a  battery,  29 
Triangle  of  e.m.f's.,  227 
impedance,  228 


U 


Unit  capacity,  34 

circuit,  158 

current,  4 

magnetic  field,  2 

magnetomotive  force,  154 

pole,  2 

potential  difference,  6 

power,  8 

quantity,  4 

reluctance,  155 

resistance,  5 
Universal  shunt,  52 


Volt,  1,  7,  8 
Voltage  sensibility,  50 
Voltammeter,  silver,  9 
Voltmeter,  calibration,  112,  114 

as  ammeter,  28 

use  of,  12 

Weston,  93 

W 

Watt.  8 

Wattless  e.m.f.,  227 
Wattmeter,    calibrated  by   stand- 
ard cell,  127,  129,  132 
comparison  with  ammeter  and 

voltmeter,  122 
Weston,  121 


INDEX  243 

Western  ammeter,  10,  92  Wheatstone  bridge,  61 
standard  cell,  9,  10  box,  66 

voltmeter,  12,  93  Work  when  a  current  moves  in  a 
wattmeter,  121  magnetic  field,  165 


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OCT  18  1952 

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